How can I see the XP amount of my assist and what is the maximum level they can get?

All Assists max out at your level. There is no XP bar for them as far as I can tell, but their level is tied to you character level.

Assist levels cannot go past a certain point until you reach specific character levels. For example, no matter how much you use them you won't get an Assist to level 5 while you are still level 1. Unfortunately I do not know the exact level break down on exactly what level you need to be to get assists to level 2, 3, etc. and haven't been able to find it. On the other hand I do know that the higher level your character is the faster your Assists level up. At level 20 you can get an Assist from level 1 to 2 within a couple of uses.

As for the effects of leveling Assists that varies based on the Assist but generally I've found you get a shorter cooldown, larger effect (whatever that effect may be), more damage, and for some (noticed this specifically on Shadowboxer and Atalan) the Assist attack changes slightly.