So I started this game a while before, left it, and then joined again.
I notice that my starter pet, Ignis, is shredded in battles with enemies around his level. He also seems to perform poorly in missions that are even below his level. He is level 59. I attuned him to the air element, and he seems to be doing slightly better, but he's still weak and having issues with missions. I'm reading about level up points, however even when he levels up I don't have the option to add more points.

Can someone help please?

Level Up points no longer exist, as they were revamped last year along with the old proficiency point system into the new current system (up-to-date guide here)

points to consider:
1. Get your second party slot at Esior's Haven Plaza
2. Unlock the second when you're in Sanctuary of Saerielle
(Currently there's the Double EXP bonus, so it's a good time to train up more pets)
Looking at Ignis, his issue...
For a start, he's an Air element pet- they tend toward being Glass Cannons, high damage output, but don't take hits as well- this is because that's what their attacks are geared towards. Air attacks are all about Damage, and offer no buffing, shielding or healing. (in the case of Air pets, you generally want Dex and Attack proficiencies)
And if he's your only party pet, that means he IS currently your tank.

For a second: His equips.
Equipment is really important in Sylestia, and he's a level 59 running three lv25 epic equips and one legendary lv25. At least his Relic and one of his bracers are myrmidon, that's the kind of equip you want on an Air pet, though not on a Tank, as they lean toward wanting Agi)
He needs an equipment update.
His relic- lv25 epic Myrmidon- has these stats:
Damage: 105 Phys
HP: 55
Mana 6
Str: 11
Int: 6
Dex: 11
Agi: 6

Here's the stats of a lv50 legendary Myrmidon relic for comparison:
Damage: 260
HP: 133
Mana: 8
Str: 27
Int: 16
Dex: 32
Agi: 16

Thank you!!!
This is super useful. Once I get more coins I will get better gear. Currently i'm also leveling up a second support pet.

Thank you! I am raising another pet right now.