Spiral Knights doesn't work like a standard RPG. Instead of XP and leveling up skills like, say, World of Warcraft, you have equipment which can be improved. This can be done in one of two ways:

1. Heating up an incomplete piece of equipment with heat embers, which are dropped from monsters. Your weapon/armour essentially stays the same, but you get more damage/defence along with extra bonuses at certain levels of heat (CTR Low for weapons at heat level 5, going up to CTR Med at heat level 10, or a +1 health bonus for armours at heat level 5; shields do not get extra bonuses for reaching higher heat levels). Heat can be transferred during a run if you revive an ally with heat, but any heat you have when your party gets on the elevator is locked to any incomplete equipment. Also, any weapon heated up to level 10 is considered complete - it can't be upgraded further with just heat.
2. Upgrading a weapon with energy, either mist or crystal. This creates a new, improved weapon or armour that may have different abilities (e.g. a Brandish gaining Fire status and split normal/elemental damage when upgraded to Fireburst Brandish; the Divine Veil gets a damage bonus versus fiends as well as curse resist and shadow defence instead of the Miracle Hood's normal defence) and a different appearance to its predecessor. The new weapon or armour has 0 heat when created, and starts over from heat level 1. Please note that this requires a recipe, as well as crowns and materials to create the new equipment, and that some equipment must be heated up before they can be upgraded to the next level (e.g. 3* equipment must have at least level 5 heat to be upgraded into 4*, and 4* equipment must be fully heated to level 10 to be upgraded into 5*).

Prestige missions give you prestige. There was a gift that was just given out, with its power depending on your prestige rank, but such an occurrence may not be repeated in the near future. As an additional minor bonus, at certain levels you also get a little badge next to your name depending on your rank, going all the way up to a horse at... what, 40k prestige? I can't remember the exact figure. They can also give you decent materials you need to upgrade your weapons - the themes are predetermined and rotate on a daily basis (beast, slime, construct, gremlin, fiend, undead), which makes it easier to find a material that drops from a certain family (say Mugs of Misery or Warp Dusts). Other than that, there isn't a great deal of purpose to the prestige missions at the current time, although there may be more benefits for high prestige later on.

Theoretically and practically, it's possible to play all parts of the game on your own. However, some situations such as bosses or boss regions may be much easier with other players. You'll probably get the most out of the game if you and your girlfriend join a guild together, since people in there should be friendly and willing to help you out if you have trouble with things.