Ever since YOTB, I've been doing numerous beefalo rushes as various characters, in solo worlds and in pubs, and would like to share my experience on the topic.
Why rush a beefalo?
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SPEED: As soon as you plop a saddle on a beefalo and mount, you may enjoy a 63.33% faster movement speed. Add on a glossamer saddle and that bumps up to 80.83%. Nobody can deny the advantage of speed, especially in the early game.
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COMBAT: Say goodbye to expensive weapons and armor because a beefalo is both! Without durability! During taming, you will be stuck with 34 damage (unless you're Wendy). But you will get a 1000 health regenerating tank from the start.
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FUN: Sick of your normal early game play? Beefalo are here for you!
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LOW COST: It costs about a dozen twigs (or anything the beefalo can eat) per day. I personally don't at all mind it because you can pick items while on a beefalo.
Character?
Some characters have perks and downsides that synergize more with others.
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Wendy, Wes, Maxwell, and Willow are fantastic with an Ornery beefalo. It seems common knowledge on the forums now that Wendy has a 54% damage boost on a beefalo if she's fighting with her sister because players lose their base damage multiplier when fighting on a Beefalo. Wes loses his health downside, and also his 0.75 damage multiplier. Maxwell loses his health downside, his shadow puppets can do things while Maxwell is riding the beefalo, and you can freely use all the nightmare fuel you don't need on an Ornery beefalo. With Willow, BERNIE! can be nice as an extra companion and tank. In fact, Willow, Wendy, and Maxwell (I know Max mains hate duelists but I think their 15 per 2 second regeneration ability is amazing) having additional fighting companions can be wonderfully helpful on a beefalo. You also don't have use for a walking cane on a beefalo so Willow can instead hold Bernie during winter.
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Wormwood is interesting. On one hand, he loses his healing downside. On the other hand, he already can sustain a perma speed boost via growth fertilizer. Up to you! Walter also loses his health/sanity downside. But he gets Woby from the start who provides similar speeds as riding a beefalo.
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I don't at all recommend taming an Ornery beefalo as Wigfrid or Wolfgang. Combat is already cheaper as them, and they lose ALL of their perks while fighting on a beefalo. HOWEVER, I do recommend taming a rider beefalo as those two. If you wanna rush one, you just have to go about it differently. I can go into that later.
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WX loses his health and speed perks. Woodie will leave his poor beefalo behind when he transforms. Wortox's healing capability doesn't work on, and isn't helpful on a beefalo. Nor can he soul hop while riding a beefalo.
Getting Started
I like to get a saddle ASAP and start taming. You'll need to find the mosaic (usually near-ish to spawn) or rockylands for at least 12 gold for science, alchemy, beefalo bell, saddle. A forest to hammer at least 2 pig houses to get 4 boards and cut stone for alchemy, and 4 pigskin for a saddle. And a beefalo savanna (I often find a road going through a little savanna near spawn leading to the big savanna) for the beefalo! I also recommend getting stuff for a football helmet and a lantern. You'll need to shave 2 beefalo at night with a razor for wool and saddle a third. Bonding with it now will mean it'll follow you everywhere, through wormholes and into the caves blah blah blah. Most importantly: it WONT go in heat and you CANT accidentally attack it. It is VITAL to bond with your beefalo before the first winter.
EDIT: After a bunch of testing and attempts to make sense of 7 separate intertwined beefalo-related lua scripts I find myself exceedingly confuzzled about what combinations of levels of domestication and bond can prevent or remove which cases of beefalo hornyness. Thank you to @gghhrr for raising this issue. I will tell you everything I think I know.
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Bonded or not, a fully domesticated beefalo will never go into heat. Not in spring nor in natural 3-day long cycles.
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If your bonded beefalo goes into heat OUTSIDE of spring, you can always fix this by entering/exiting the caves with the bell.
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My bonded beefalo have not been going into heat in the first winter. It may be because they are already mostly domesticated.
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It does seem like beefalo that are not yet domesticated will unavoidably go into heat in spring, bonded or not.
EDIT: Apparently a bell does not currently prevent you from accidentally attacking an unsaddled beefalo ![:wilson_cry:]( ":wilson_cry:")
See my bug report for details.
The first time you mount the beefalo, it will buck you off in a measly 15 seconds. Fear not, however, as this quickly increases over time.
When to feed the beefalo
The beefalo's domestication WILL be increasing as long is it's being ridden. Keeping it's "hunger" stat (300 hunger per day) up does not also increase domestication if it's being ridden, so don't bother. For your beefalo to allow you to ride it, you need to maintain the "obedience" stat. Feeding a beefalo anything grants 10% obedience. Expect obedience to decay at a rate of 100-150% per day. When obedience goes below 50% your beefalo will not let you back on. So feed it 5 things and remount. When obedience goes below 40% it can shake off it's saddle, decreasing it's durability. Feed your boof at least 6 things, put the saddle back on, and mount it. Note that obedience will not be an issue once the beefalo is fully tamed.
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The simplest way to know when your beefalo needs to be fed is to just feed it 5 things when it doesn't let you back on. The downside is that the beefalo will also often shake off it's saddle. You must be prepared to craft extra saddles with 4 gold, beefalo wool, and pigskin.
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I usually just eyeball it and feed my beefalo when I dismount. This means I probably waste a couple grass or twigs every now and then by feeding it when I don't need to, but I don't mind.
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A lot of people use a mod such as "show me" that displays information such as the obedience stat.
But what if I need to do things off the beefalo?
First of all, don't worry about salt licks. Domestication loss starts off negligible and you shouldn't need to leave your beefalo unmounted for too long. But when you do, there are 2 options.
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It'll only increase the total domestication time by a little. Don't worry about it.
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If you're obsessive about trying to tame your beefalo as fast as possible without delay like me, keep it fed while you're not riding it. Each twig/berry will satiate it for 15 seconds and a carrot/juicy berry for almost 20 seconds.
Glossamer and War Saddles
Sadly it's difficult to get the war saddle early when ewecus chances are low. It costs 4 steel wool and a ewecus drops 2-3. It increases damage by 16 so a war saddled, ornery beefalo does 66 damage. I definitely recommend making this one with a construction amulet once you can. The glossamer saddle costs 68 butterfly wings, 4 silk, and 2 living logs. Might seem like a lot but this is doable even without an amulet early on. If you're playing with others I suggest farming butterflies at base with other players nearby to increase butterfly spawns. I might also recommend a quick little ruins rush with your beefalo for a construction amulet if you're up for it.
Quick Ruins Trip is Totally Doable!
This is in no way needed obviously. I've been experimenting with making a quick trip to the ruins for just a construction amulet for future glossamer and war saddles, a couple star caller staves, and maybe some frazzled wires or a crown. All you really need is a lantern, tools, a football helmet, maybe some blue caps to be safe, and maybe moggles if you really lack confidence but I find the beefalo to be fast enough that you can run right past clockworks without a problem. You can feed beefalo light bulbs by the way. Avoid fighting clockworks; just dodge them. Mine some statues, I like to run around the rook set pieces (stay away from the bishop), and craft some things. I suggest going later in autumn so the buck timer is longer but you do still need to take opportunities to dismount and remount! It might be a good idea to farm 34 butterflies on days 9, 10, and 11, do what you need to do on the full moon, and then go on day 12 with a precrafted alchemy engine to make the glossamer saddle as soon as you get the green amulet (only needs a broken station to craft). I was able to get 34 butterfly wings on days 18, 19, and 20 before winter hit but it was tight.
Tendencies
The ornery tendency is ideal for rushing because you're free to fight while riding. And a base of 50 damage once tamed is very valuable. It's downside is that it has a minimum obedience of only 45% once tamed, meaning you will often need to feed it 1 item to mount it. The other tendencies have a minimum obedience of above 50% so obedience no longer has any effect. If you wanna tame a rider beefalo (it is about 15% faster, without ornery downside, base 25 dmg), I suggest to not worry about taming it quickly and just use it to get around. I personally haven't tamed default before because I don't mind the downside of Ornery. You can tell it's tendency by it's expression. For ornery you just need to attack a lot and/or get hit a lot. For rider just don't do those things. You shouldn't have much trouble.
When will my boof become fully tamed?
It takes a minimum of 20 days after you start to complete taming. Once you do, the buck timer will skyrocket and you'll gain benefits of the tendency you trained. Definitely expect more than 20 days. Depending on your goals, you can be more or less careful to keep domestication rising. You'll probably want a fully tamed ornery beefalo for when deerclops comes on day 30, but that gives you plenty of time.
Other notes
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If you can, make a piggyback because it's speed penalty doesn't apply while you're riding a beefalo.
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PLEASE dismount and remount your beefalo before combat to reset the buck timer.
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Feel free to ask any questions or comment on your beefalo rushing experience
Perhaps Klei will take a few things from this and make something of it. My honest opinion, I think its a gate to the Moon or Hamlet or a combination of both. Because everything in the Archives tells us its from the Moon or "These people loved the moon" as well as the markings resemble that of Hamlet. 
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