The three saving throws are fortitude, will and reflex. There are many ways to increase these but it is always best to start by increasing the base save. The base save is calculated by simply adding up the individual saves of the classes taken pre-epic. After epic levels have been reached all saving throws will be calculated the same way.
A class' primary save is calculated as
[Class level / 2 ] + 2 (rounded down) For example a fighter's base fortitude save at level 17 will be: 17 / 2 = 8.5, 8.5 + 2 = 10.5, 10.5 rounded down = 10.
A class' secondary save is calculated as
[Class level / 3 ] (rounded down) For example a fighter's base will save at level 17 will be: 17 / 3 = 5.66, 5.66 rounded down = 5.
All epic saves are calculated as
[Every epic character level / 2] (rounded down), added to the saves worked out for level 20. For example if a character's saves at 20 were fortitude 10, reflex 8, will 8, then at level 31 the saves would each be increased by 11 / 2 = 5.5, rounded down to 5: fortitude 10+5=15, reflex 8+5=13, will 8+5=13.
Thus, the base saves at level 40 are 10 more than the base saves at level 20, regardless of which classes are advanced in the epic character levels.
This means that combining three classes all with the same primary save gives a higher base save than with one class. It also means in order to get the best primary saves possible, all class levels should be kept as multiples of 2. Also, in order to keep secondary saves as high as possible, class levels should be kept as multiples of 3.
A level 20 paladin will have the following base saves at level 20: fortitude 12, reflex 6, will 6.
A level 12 paladin, level 4 champion of Torm, and level 4 monk will end up with the following base saves at level 20: fortitude 8+4+4=16, reflex 4+4+4=12, will 4+1+4=9.
Attack bonus is another consideration. However examples 2 and 3 above could both be developed into a a 20/10/10 paladin / champion of Torm / monk, and both have a base attack of 19 at level 20, yet example 3 ends up with better reflex and will saves.
All saves are also affected by ability modifiers. Fortitude is modified by constitution, will by wisdom, and reflex by dexterity. Many classes will almost always have higher saves due to particular stats that are usually high, such as cleric with high wisdom.
Saving throws can also be increased magically. However a cap of +20 exists on this. The cap applies to bonuses from magical equipment, spells, spell-like feats (e.g. bard song, divine wrath) and the feat sacred defense.
There are some feats that increase saves against only against some effects like resist disease which only gives +4 to fortitude saves against disease that have not been included.
First level feats
Class/race specific feats
Here are list of all the saves and the classes that have them as a primary save