I'm playing a protector aasimar paladin and I was wondering how I can get more spell slots.
By advancing your character's level
The only way to increase your total amount of spell slots is gain more levels. You can either continue gaining levels in paladin or you can multiclass into another class. If you do multiclass, please see this chart which details how your spell slots would work.
Paladins are known as 1/2 casters, so their spell progression isn't as big as full casters. The trade off in multiclassing to get more slots from a larger caster class will be a delay in your paladin progression (which is a big deal, paladins get a lot of good stuff!)
It is important to note that the multiclass rules are entirely optional and up to the DM as to whether or not you are allowed to use them.
While gaining levels or multiclassing give the character more slots, there are ways to reuse slots.
For instance, the Pearl of Power:
While this pearl is on your person, you can use an action to speak its command word and regain one expended spell slot.
Any full caster or a Warlock (depending on the usual number of short rests) will be a big boost.
Sorcerer or Bard
1 level will not help, but any more will give you a lot of extra slots.
Depending on the number of short rests you usually get in a day, it might be better or worse than Sorcerer.
Comparison to single-class Paladin
The most important (for DPR) Paladin feature is Extra Attack, the second is Divine Smite, the third Improved Divine Smite.
Adventuring days are short 1, so IDS adds about 10d8 damage to your days (if you win all initiatives, less if not). If you go Paladin 8/Sorcerer 4 instead of Paladin 12, you get (3 * 4 + 2 * 3) = 18 d8 from the extra spell slots in case you spend them for Divine Smite. You are way ahead on raw damage, and in my opinion, the spells add much more versatility than what you lose on Paladin class features.
Paladin levels 5, 6, 8 are all good places to get out.
1) I have participated in nearly 150 sessions of 5e, and no encounter lasted longer than 5 rounds, on average a bit less than 3 rounds.
One adventuring day on average had no more than 5 rounds.
Data aggregated from 5 different DM's.