I like that Mountain Druid kit, although I can't take a kit when multiclassing, I think. Maybe I can take a few levels of fighter and then dualclass to mountain druid. Or just take fighter/druid multiclass.
15 cha and 12 wis as required stats, though ? Eh.
Rolling stats, 4d6 - lowest... 14, 13, 12, 16, 14, 13... Don't have to complain.
I don't get this edition much...
Your class would be Fighter / Mountain Druid. The Mountain Druid thing is effectively replacing your normal Druid class abilities. You would then have a kit layered on your character like a plastic transparency sheet. A kit doesn't change your class.
I think, though I haven't looked for it in the text in some time, that kits work like this:
1: You can have no kit, or one kit.
2: You must have all of the classes a kit requires.
3: If you have extra classes that's fine, but you can't take another kit to cover those extra classes.
So for example, if you're a Fighter/Thief, you can take a Thief kit, or a Fighter/Thief kit, but you cannot take a Cleric kit (because you don't have any Cleric) and you can't take a Fighter/Mage kit (because you lack Mage).
Normally, only single-class Fighters can specialize in a weapon. But some Fighter kits give you a free specialization as their benefit (Myrmidon?). That means you can be a Fighter/Thief, take the kit, and get a free specialization. But of course that doesn't let you improve the skill to double-specialized which is a nice thing to have.
The classes fit into class groups. Fighters, Rangers, and Paladins are all "Warriors". Clerics and Druids are all "Priests".
Check the beginning of the kit chapter in the book you're looking at. In the Complete Fighter's Handbook, for example, it explicitly describes these kits as usable by any "Warrior" classes. Which means you can take a Paladin, give him the Myrmidon kit, and get a single free weapon specialization. Or you could take one of the kits from the Complete Priest's Handbook, which says it can apply to any Priest - including the Cleric, the Druid, or one of the specialty priest classes described in the book. Those Priest classes are kinda like your Druid handbook branches - they actually replace the class so you would be a Fighter / Priest of Nature for example. This kind of direct class alteration is sort of rare.
It sounds like you understand dual-classing, but for everyone else: multi-classing is when you have more than one class at the same time. Dual-classing is NOT just multi-classing with two classes. It is a special thing Humans can do, where they start in a single class and then later abandon it to take up a different class. The ability score requirements are intense and you don't earn new HP and stuff until you surpass your old level, so there are limited instances in which it's a good idea. But it's a possibility. They really should have called it something like "Class-Changing" instead.
Furthermore, while my group always allowed a "where you're from" kit (Warriors and Rogues of the Realms, etc.) in addition to a class kit, I suspect you're actually supposed to take a "homeland" kit instead of a class kit.
All that is just stupid stuff that's not entirely important; as long as you expect that your multi-class character isn't going to be quite as good of a Fighter as a single-Fighter, and not quite as good of a Druid as a single-Druid, you're fine. If you choose Mountain Druid then your class would become Fighter / Mountain Druid, and you can pick a kit that you qualify for.
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How multi-classing works is you need to take all those classes at once, at first level. So you'd be Fighter 1 / Druid 1. As you gain EXP you split it up among all your classes. Meaning if we earn 4,000 XP each, you would have 2,000 in Fighter and 2,000 in Druid. You train for each class separately, paying training costs as Rem demands, etc.
When you go up a level, you roll HP and then split it by your number of classes. Then add CON modifiers. So a F1 / D1 would roll d10 for F and d8 for D, halving each die, adding together. Let's say you roll 6 on d10 for Ftr and 5 on d8 for Druid. That would halve to 3 for Ftr and 2.5 for Druid. You'd round Druid (the rules of mathematics say when you round to the nearest whole number, you round anything under 0.5 down and anything 0.5 or over up - Rem may say you have to round down or up or whatever) to 3, so 3 for Ftr and 3 for Druid = 6 HP at level 1/1.
If you gain a level in Ftr first, you'd roll d10, split in half, add to your total. Then when you level up in Druid you do the same for the Druid's d8 hit points.
But your saving throws are the best in each category for your classes. That probably means you'll use the Priest group saves (as a Druid) for most of it because Warriors (as a Fighter) start out with weak saves.
Typically when you multi-class you average or use the better of the two, depending on what you're looking at. But class abilities are based on the level of that class, so your Druid casting level will always be your Druid level regardless of what Fighter is at. It's not additive like 3E.
Because of how the XP tables work, you'll probably find that a multi-class character is only one level behind a single-class, until "name" level around 9 or 10 when the single-class characters surge ahead. That is, if the party has a Fighter at level 5, the party Fighter/Thief will probably be 4/4.
(Although, some classes have strange advancement rates. Druids and M-Us do some weird stuff at different places.)