So, I'm making a Devotion paladin. What spell slot level would I use for, say, Lesser Restoration (which is received at level 5)?
The spell description tells you what the spell’s level is.
Lesser restoration is a 2nd level spell, as stated in the spell description. Every spell description tells you what level it is. Here is a picture of the spell description from page 255 of the Player’s Handbook (also here on D&D Beyond, and page 99 of the Basic Rules pdf):
As you can see, the spell’s level is listed immediately after the name of the spell. Since the Paladin’s class description doesn’t tell you any information about your oath spells besides their names, you will have to consult the spell descriptions in the PHB (or other books if using an oath from another book giving spells from another book).
First, you need to understand the section on Spellcasting.
You have spells that you have prepared. And you have spell slots. Think of these two things separately.
Spells Prepared As a paladin, you generally have a number of spells prepared equal to Charisma modifier + half your paladin level. So as a 6th level Paladin with 14 Wisdom you could prepare 5 spells (3 [half-level] + 2 [Wis mod]). The highest level spell you can prepare is a 2nd level spell because that's the highest level slot you get.
Spell Slots And then you have your spell slots: Looking at the Paladin table you get 4 1st level slots and 2 2nd-level slots.
General Spellcasting Rule Now, hopefully you are aware that whenever you cast a spell that you have prepared you 'spend' a spell slot to cast a spell of the slot level or lower. So to cast a 2nd level spell you must spend a spell slot of at least 2nd level.
Devotion Spells Now we get to your devotion spells. These are additional spells that you always have prepared and don't count against the normal number you can have prepared. Note this has nothing to do with your spell slots.
So as a 6th level Oath of Protection paladin you would get the spells Protection from Evil and Good, Sanctuary, Lesser Restoration and Zone of Truth. These additional 4 spells would always be prepared in addition to the 5 mentioned above. So you would actually have 9 spells prepared!
The rules for casting these additional spells are no different to cast any other spell. When you want to cast one of them you spend a spell slot of at least that spell level or higher. So to cast Lesser Restoration - a 2nd level spell - you must spend one of your 2nd level slots. (You can spend a higher level slot if you want, but this may or may not 'heighten' the spell - you need to look in the spell description: some spells do more if you use a higher level slot to cast them).