In the Basic Rules for D&D 5e, the number of spells a wizard can prepare is somewhat determined by your ability score, but it seems that the number of spell slots is constant, no matter what your ability. Is there a way to increase the number of spell slots a wizard can use? (Aside from leveling up, of course.)
At this point, there is no way by RAW in the Basic Rules (or in the Player's Handbook) of increasing your number of spell slots you have as a Wizard.
That said, there are a few ways of effectively increasing your number of spell slots - either by restoring spell slots or providing other ways to cast your known/prepared spells - in the Dungeon Master's Guide using either Magic Items or the optional rule of Epic Boons (DMG, p.231-232).
Some examples of Magic Items that provide this benefit include:
- Pearl of Power (DMG, p.184)
- Rod of Absorption (DMG, p.195)
I don't know of a way to increase the number of actual spell slots you have as a pure wizard, but you can make a trade off by multiclassing.
When you gain levels as a full caster (bard, cleric, druid, sorcerer, wizard), the levels all add together to determine the number of available spell slots, which are shared between them. That is, a wizard 5 will get 4 first, 3 second, and 2 third level spell slots. A bard 1/cleric 1/druid 1/sorcerer 1/wizard 1 will get the exact same spell slots, but a very different spell selection.
If you take a two-level dip in Sorcerer, you get two features: Sorcery Points and Flexible Casting. You can use the two points afforded to you to create a shiny, new level 1 spell slot, usable for any level 1 spell you know (wizard or sorcerer).
The biggest downside is that you'll always be two levels behind in Wizard spells and prepared, meaning you will only know level 3 Wizard spells when your compatriots know level 4 spells. You will still be able to cast spells in level 4 slots for more effect.
Another tradeoff is you'll also permanently have prepared three level 1 sorcerer spells (again castable with the shared, possibly higher, slots) and four cantrips. However, you'll again be down two Wizard levels meaning two prepared spells from your book. You net one spell, but only if you were going to prepare those exact, fixed sorcerer spells (and cast them with Charisma).
Well... technically, you can craft scrolls for any spell you know without spending a spell slot. It is single-use but it doesn't consume a spell slot, effectively giving you a free xth level spell. So each scroll is in effect a reserved but free spell slot. Have fun filling your bag of holding with a hundred or so scrolls. Also, enchant a longsword if you are an elf, because enchanted weapons are the best. Besides that, you can probably talk your DM into putting a few spells a la spell scroll onto your staff or something, since spell scrolls used to be multiple-choice. Good luck.