The maximum number of artes you can do in a combo is determined by the number of Souls in your Soul Gauge (filled diamonds), e.g. if you have 2, you can only do 2 artes. Performing a Break Soul (R2) will consume 1 soul. You can also lose souls by being stunned or hit by certain status ailments.
You can earn Souls in a variety of ways: killing enemies, stunning enemies, perfect dodging (periodically), and some food effects.
Look at your artes for moves with the Stun property on them. For Velvet, that's most of them. Chance to stun goes up the more souls the defending target has, and the fewer attacker has.
I've read that once you get a potentite from an arena or something, you get the ability to hold a shoulder button down for access to more artes, but I'm 20 hours in and haven't found that yet. But, welcome to Taels games! In Vesperia I think you could have access to 8 artes at once. Now you technically have access to 16 from the start.