The correct answer is that it depended on the DM
Any OD&D question that covers the period from 1974 to 1977 has to account for DM choices, or the question is quite simply wrong in its assumptions. If you wanted the "official answer" but nobody in your group got the Strategic Review, for example, you didn't even know that the Ranger or Bard class even existed. Or the Illusionist. But to address your question ...
If you are a player purchasing Dungeons and Dragons rules in order to improve your situation in an existing campaign, you will find that
there is a great advantage in knowing what is herein. If your referee {DM} has made changes in the rules and/or tables, simply note them {underlined} in pencil {underlined} (for who knows when some flux of the cosmos will make things shift once again!), and keep the rules nearby as you play. A quick check of some rule or table may bring hidden treasure or save you fame "life". (Men and Magic, p. 4)
First of all, the game was not crippled by RAW Psychosis, as it has since become burdened, and as it became during the AD&D 1e "attempts at standardization" era that building modules for conventions brought with it. (Modules were a revenue stream). I think that trying to answer the torrent of questions that descended on TSR back then (via mail or in Dragon Magazine) led to some of this. Gygax's "we can write a rule for that" approach ran amok, to put it mildly, but a lot of what led to that was fans and players who kept on asking, via cards and letters and questions in the Dragon Magazine, or at cons, "but what about this?" fed a need to find an answer. TSR tried in many cases to answer them all - but not at the speed of the internet. We didn't have the internet. There was no standardization: there were guidelines, not canon, not an official way to play. (A lot of best practices were discussed in Strategic Review and in Dragon Magazine, however).
Touch tone phones are the exception, not the norm. We dialed to call. We'd write questions in letters in self-stamped, self-addressed letters to TSR. TSR's small crew of folks, bless them for this, did their best to answer what came in.
The president of our college gaming club did indeed do as was suggested on the title page to Men and Magic. He wrote in a question to: Tactical Studies Rules, POB 756, Lake Geneva Wisconsin 53147, about a problem he was having (we were having, due to Demons in Eldritch Wizardry) with understanding magic resistance. He never heard back, or, since he wrote in the spring of 1977 and graduated in May of 1977, the reply may have gotten to him but we'll never know.
Secondly, each table's DM made differing decisions on that as the DM deemed suitable. That's how the game was played, back then.
How we did it: Case 1
At most of the tables where I played: once you reached the next level, be it Fighter, Thief, Magic User, whatever with a multi-classed character (Dwarf Fighter Thief, Halfling/Hobbit Thief/Fighter, Elf Fighter Magic User, Half Elf Fighter Thief Magic User (per Greyhawk)) you rolled the next HD. That added an amount to your HP.
Example: the Dwarf Fighter Thief, when he got his second thief level, rolled a d4 and added that to his HP on top of the original d8 and the original d4 for the F1/T1 he was at the time. (Obviously, this is with Greyhawk included).
How we did it: Case 2.
At the start of each adventure, the DM rolled your HP based on the HD you had, and then told you how you felt. A lot of the DMs I played with did that at the start of the adventure session. All HD that a character had earned at that point in their career were rolled.
Example a 2/2 Fighter Magic User elf had rolled 2d4 and 2d8 if you were using Greyhawk, or the d6 and mods if we were not using Greyhawk. Those were your HP. (+ con mods, if there were any).
How we did it: Case 3
The players did per Case 2, based on the adventure that day for that session. That means that my Greyhawk era thief, level 3, would have rolled 3d6 (he had a Con of 10) before each session. I knew if I was having a robust day or a sickly day.
If this sounds odd, to the modern D&D player, consider this: the DM rolled the HP of each monster as we encountered them during random encounters, which were plentiful. Us rolling the HD made sense within that context.
Your questions are, based on my experience, somewhat at odds with how it was done at the table where I played. To a certain extent, you are engaging in an anachronism here in trying to shoe horn modern understanding of the game (standardization) to its original and very sparse rule set. I am sure that someone who played at Gary Gygax's table would answer this differently, and someone playing at Jim Ward's table might offer a different take.
I didn't play in the Twin Cities, nor the Lake Geneva area. I played mostly in Maryland and Virginia, in high school and college. We had no concept of "there is an officially sanctioned way how to do this, thou art a vile sinner if thou doest that not!" We worked with what we had, and played. Each table run by each DM was a bit different from the others. That was, for us, part of the fun. Chris's dungeon was different from Tom's dungeon was different from John's dungeon was different from Richard's dungeon was different from Mike's dungeon.
It was a big freaking deal when Supplement 2, Blackmoor, showed up at the local hobby store. I was lucky to get a copy before they sold out.
Were there other ways of dealing with multiclassing? Sure. Each DM dealt with it as it came up, wherever they were. (per Men and Magic page 4 citation at beginning of this answer).
Bottom Line
We did it however the DM ruled it. That's how we did it back then. Somehow, we had fun. 😁
For XP
Do you earn XP separately for each class? i.e if you earned 2500 XP as a Fighting-Man, does this mean you can play as a second level Magic-User when you make the change, or do you start the new class
with 0 XP?
Neither, in my experience. Not sure how other tables did it. The way we did it was to split the XP evenly between classes once the second class was identified. However, I do not recall any of our players playing an Elf Fighter Magic user before we added Greyhawk supplement to our standard games (all of us) once it came out. Simply surviving level 1 was a cause for celebration. The most common scheme I saw then was that if you wanted to be multiple classes as an elf, the DM would have you split XP evenly from the beginning. Page 5 in Greyhawk was only partly helpful as a guide, and the way I saw it done was certainly a variation of the strict interpretation of the passage in Men and Magic (which would imply the latter, not the former, status. Getting to level 2 as an elf would require 4500 XP to become both a level 2 F and level 2 MU).
How should I do it now? I want to play OD&D in 2021!
However you want to. Arrive at whichever scheme makes the most sense at your table. Talk to each other, and arrive at what makes the most sense to you.
Or, and I say this because I have great respect for one of our contributors, post a question to Delta'sD&D Hotspot. His take on Original D&D is pretty solid. (The blog author contributes here now and again).
