The rules aren’t actually explicit about how this works; they say only that
Base Attack Bonus
Add the base attack bonuses acquired for each class to get the character’s base attack bonus. A resulting value of +6 or higher provides the character with multiple attacks.
Saving Throws
Add the base save bonuses for each class together.
The issue with this is that characters can gain BAB and base saving throws in fractional amounts per level:
These have to be rounded down, since a fractional bonus isn’t a thing in these systems. For single-classed characters, they are rounded down in the table. For multiclassed characters, these figures could be rounded down first, for each class, and then added together, or they could be rounded down only after adding all the characters together. The core rules do not specify an approach.
Sum of Rounded Values: From the Table
This is what many groups view as the “default,” even though the core rules don’t actually specify. Since the numbers in the classes’ tables are all rounded down, it’s easy to just read those numbers and add them.
The problem with this is, you end up with really weird saving throw numbers. Take a 4th-level paladin/8th-level barbarian/8th-level fighter: they have base saving throws of +16 Fortitude, and +3 Reflex and Will. A single-classed character of any one of those classes would have +12 Fortitude, and +6 Reflex and Will.
Rounded Sum of Values: Unearthed Arcana “Fractional” BAB and saves
The opposite approach is to add first, before rounding. Since the un-rounded numbers aren’t directly printed in the book (the way the rounded numbers are printed in classes’ tables), many tables consider this a “variant,” a belief borne out by D&D 3.5e’s Unearthed Arcana and Pathfinder’s Pathfinder Unchained using that term. However, again, the core rules don’t exactly come out and say the other approach is official.
Variant or not, though, this approach is highly recommended.
The idea here is to simply round after you add the classes’ BAB and saves together. The means knowing the un-rounded values of those numbers, so there’s some moderate extra work involved, but you get much more appropriate results.
With the Unearthed Arcana “fractional saves” approach, that 4th-level paladin/8th-level fighter/8th-level barbarian still has a Fortitude save of +16, but has the same +6 Reflex and Will as the single-classed characters of those classes.
Skip the Repeated +2: A Common 3.5e Houserule, and Unchained Suggestion
Unearthed Arcana does not suggest this, but it seems so obvious that many D&D 3.5e players just assume it does, and it is therefore an extremely common houserule. Pathfinder Unchained includes it. The issue you’ll note about the paladin/fighter/barbarian above is that they have the same Reflex and Will saves as a single-classed character, but considerably higher Fortitude save. This is a problem.
It’s caused by the fact that “good” saves are +2+½×level—that +2 gets repeated for each class that has that good save. To avoid it, just don’t repeat the +2. This has the happy side-effect of making the math easier: you just add up the number of “good” levels you have and the number of “poor” levels you have, and then only need to perform at most one addition per save, no matter how many classes are involved. For many characters—including our paladin/fighter/barbarian—you won’t even need that, since if you have all-good or all-poor saves, you just need to look up the single-classed value for a good or poor save at that level. Under this rule, a paladin/fighter/barbarian will always have exactly the same base saving throws as a single-classed paladin, fighter, or barbarian. No math needed at all!
Bonus Saving Throw Feats: A Less-Common Houserule
This isn’t something I’ve seen at other tables, but it’s something I’ve done at mine and I really like it: instead of good saves coming with a +2, they come with Great Fortitude, Lightning Reflexes, and/or Iron Will as a bonus feat. This provides a +2 bonus that doesn’t stack with itself, so it works out the same way as skipping the repeated +2, but it also grants characters common feat taxes for free. This is a very good thing, as feat taxes are terrible in these systems and these are very, very poor feats indeed to get taxed with.
Including Prestige Classes
For prestige classes, the save progression is also either good or bad, but in Pathfinder for whatever reason the progression has been shifted (and good saves lose the +2). Under the fractional system, these changes don’t make sense and should be ignored (even under the default system, they make limited sense). Just note if a save is good or poor, and treat the prestige class levels as levels of good or poor save progressions, as normal. Note that “good” saves start at +0 while “poor” saves start at +1, because this shifting was poorly-considered and shifts good and poor saves differently.
Conclusion
I typically strongly suggest both fractional and no-repeated-+2 for games I play in. Games I run, I use the bonus feats instead of “just” not repeating the +2. These result in everyone having base saves that are roughly “level appropriate” regardless of the timing of any multiclassing.
You are correct. Spell saves are 10 + spell level + spellcasting stat modifier (+ any miscellaneous bonuses). They don't go up directly with character level, except insasmuch as character level lets you cast higher level spells, get higher stats, and get feat like Spell Focus that boost your DCs.
As you become a higher level caster you need to either start relying on spells that don't allow saves as much, target opponents' weak saves, and/or double down on your spellcasting stat modifier and feats to get those DCs up. Stat boosting items or spells help, as do a variety of specialized +1 DC items.
Tactically you'll find that combining save spells with other debuffs (from yourself or other party members) will be necessary as you reach high levels (10+).
Best Answer
Yes. Each level selected provides cumulative benefits.
This also applies to skill points and base attack bonus. See the section on multi-classing in the PHB or the SRD (under classes) for details. As you are selecting base classes, there are usually no prerequisites (see monk and paladin for exceptions).
Please note that when leveling up and selecting prestige classes, those may have prerequisites which are required to be met before taking that level. There are other small details, but it does not seem relevant to your question, so I will avoid expounding.