I'll be playing a Paladin for the first time in an upcoming campaign (dungeon of mad mage). Not much of a surprise but with the makeup of the party, they will rely on my character to deal heavy damage. Having never played a Paladin before I wouldn't mind some suggestions for spells.

This is the info I have set in stone, either because of the setting or the backstory I have:  Level 5 Human variant. Oath of Vengeance. Sentinel feat. Polearm master feat. NOTE- I'm still not set on a fighting style so would appreciate any thoughts on that.

So at 5th level paladin:  (4) first level spell slots (2) second level slots. Appreciate any advice

The way to deal damage as a pally is to more or less ignore your spells. Just use the slots to smite, especially on an crit, and you’ll be putting up big numbers. 
The smite spells are mostly trap options. There’s a couple that can come in useful situationally, but by and large, ignore them and just do basic divine smite. Normally I’d say keep find steed prepared, but in a dungeon, it usually won’t do you much good. 
That said, I’d take misty step because it’s always useful. You might want shield of faith, but really, every spell you cast means one fewer use of smite, so it’s a tricky balance. 
For style, I’m guessing you’re going heavy weapon, so I’d pick defense — you can’t do damage when you’re down. 

Divine favor is not a bad spell either for a consistent damage boost. I mean, adding 2d8 on a single attack is nice, but adding 1d4 on every attack for up to a minute for the same spell slot is sometimes nicer. Even if it only lasts for two rounds, after 5th level you’re still statistically a teeny bit better than that 1 divine smite. That doesn’t include bonus action or reaction attacks from PAM/Sentinel either, which put divine favor over for total additional damage per spell slot. Every round after that is gravy on top. If you’re fighting a hoard of mooks then divine favor is far more useful than a single Divine Smite, Divine Smite is often better against a single big hitter.

And compelled duel can come in clutch at times too. You won’t use it often, but when you do need it you’ll want it prepared.

bless is underrated and I'm not really sure why. It does some WORK for you. The bonus to hit alone is worth it but the bonus to saves is amazing too. Hitting more often means smiting more often. 

The craziest part of bless is it doesn't benefit just you. 2 other party members get the perks too. 

Combo bless, PAM, and GWM with a vengeance Paladin rocking their Vow of Enmity and your damage potential is just nutty. 

Sure, it is always better if you got someone else blessing you than you doing it yourself. You could absolutely benefit from that, since you got other spells you could be rockin. Divine Favor, Hunter's Mark, both okay options. And Shield of Faith is decent for defense. 

Yeah, I’d put bless up there with divine favor and shield of faith, I kinda figured it was so good it wasn’t necessary to call it out. Who underrates bless?!? Bane gets underrated all the time, but bless? Really?