Settle an argument for me please.
This is a stupid question, and I can’t imagine any scenario where a player/manager would ever do this, but if a batter is hit by a pitch, does he have to go to first? I mean is there an option to shrug it off and continue batting, or is it mandatory that he go to first?
Has to go.
A better question is why the guy who just got hit becomes a batsman, not a batter. 
Well if he knows it will hit him he can swing and miss or try and convince the umpire in some way that he intentionally walked into the pitch. In either of those cases he would not be eligible to take first.
The first example happened in the Pirates/Diamondbacks game two days ago.
He has to go to first. From the official rules:
6.08 The batter becomes a runner and is entitled to first base without liability to be put out (provided he advances to and touches first base) when -
. . .
(b) He is touched by a pitched ball which he is not attempting to hit. . .
6.04 A batter has legally completed his time at bat when he is put out or becomes a runner.
So he can’t remain at bat because he has become a runner. I can’t find anything in the rulebook covering the case where the batter-runner refuses to go to first after being hit by a pitch. The rule about abandoning the basepaths only applies after the runner has reached first. One possibility is that the home plate umpire would declare the batter out under rule 9.01 ©:
“Each umpire has authority to rule on any point not specifically covered in these rules.”
The only other resolution I can think of would be for the umpire to eject the batter-runner and allow the manager of the offensive team to put in another player as a substitute. This is what happens if the batter-runner is ejected for charging the mound after being hit by a pitch.
In order to be awarded first base, however, I believe the batter must have attempted to avoid being hit, if at all possible.
Also, you might enjoy this; you’ll see its relevance to the topic at the very end. 
Why is there such a rule in baseball anyway?
You’re asking, why is the hit-by-pitch, advance-to-first rule a part of baseball?
Because it’s not dodgeball, and any pitcher who hits a batter with the ball should be penalized. Not to mention that if it hits in a particularly tender spot, it would hinder the batter’s ability to continue the at-bat.
I’ve always wondered whether a batter has to circle the bases and score after hitting a ball over the fence for a home run. Let’s say he’s already hit a single, double, and home run in the game, and he decides he’d like to hit for the cycle. Can he stop at third base for a triple?
He’d probably get thrown out of the game - disrespecting the spirit of the game. I have no idea if he’d get credit for a triple.
If he fails to touch home, he would in fact get credit for a triple.
Yes, he’d probably also be ejected unless he made it look like an accident; the usual term is “making a travesty of the game.”
Never knew that was an ejectable offense. As a life-long Phillies fan, I’ve seen a number of years (much of the 80s and 90s, for example) where the whole team could have been ejected for this.
Also, I’m chuckling over the thought of team with a dugout on the 3rd side. Guy who is a triple short of a cycle puts the ball over the fence, runs to 1st, 2nd, and 3rd, then keeps going into the dugout. He’s out for leaving the baseline, no?
No, technically he’s out for “abandoning his effort to touch the next base.”
The rule against just leaving the basepath applies when you’re avoiding a tag.
This reminds me of a joke. A pitcher throws to a batter, and his very first pitch is drilled for a double. The pitcher manages to do well the rest of the game, though, until the same batter comes up to the plate.
On the guy’s second AB, he hits a triple. The pitcher again recovers and stays in the game, and faces the guy again.
On the guy’s third AB, he knocks the ball completely out of the stadium for a home run.
Well, when it comes time for the guy’s fourth AB late in the game, the manager-- realizing this batter has his ace’s number tonight-- finally pulls his starter and brings in a relief pitcher.
Accepting the ball from the starter, the reliever asks, “Any tips about this guy?”
“Yeah,” the starter replies ruefully. “He can’t hit singles.”
If you really want a triple rather than a home run say to hit for the cycle, you can do so easily providing there’s a guy on base. Just tell the guys in front of you that if you hit one out and they’re on base, that one of them has to stop between third and home. Then when you come around third, pass the stalled runner and be called out for passing him. Voila- home run becomes triple.
Yep. This came into play when Don Drysdale was in the middle of his then-record streak of 58 consecutive scoreless innings. When the streak stood at 44 innings, Drysdale was pitching with the bases loaded when he hit Dick Dietz with a pitch. This should have forced home a runner and ended the streak. However, the umpire ruled that Dietz didn’t try to get out of the way and ordered him back to the plate. Given another chance, Drysdale was able to get out of the inning without a run scoring and was able to continue the streak.
Zev Steinhardt
There are certainly such scenarios where the batting team would prefer to have the batter at the plate rather than awarded first. But they rarely play out, because that’s the exact scenario where the pitching team will give an intentional walk.
So you don’t really need to imagine very hard, just look for intentional walks.
I assume there’s some kind of sabremetric study of how often the pitching team is actually (statistically) correct in awarding intentional walks?