From my short experience (I've been in a poly relationship for almost six months) the skills I've needed in my emotional toolbox so far include:
- Ability to ask for what you want and need from your partners, hear what they want and need, and then negotiate. You can't unilaterally demand what you want, nor can you just yield more than you're really okay with in order to avoid conflict. It's got to be a very open give-and-take.
- Ability to acknowledge negative feelings (like jealousy, insecurity, resentment) and work through them. Everybody experiences them; if you (like me) are the kind of person who naturally stuffs them down and pretends they're not there, the stresses of a poly relationship are pretty much guaranteed to trigger an explosion sooner or later.
- Security in your current relationship. You have to trust that your bond is strong enough to weather some tough times. If you're not there yet, work on the relationship between the two of you before bringing someone else in.
- Good boundaries. Ability to distinguish between "my problem," "your problem," and "our problem." Ability to enjoy and invest in activities that don't involve your partner, for your partner to do the same, and for you both to recognize that that's healthy and not a threat to your relationship.
- Trust that your partners also have the above skills.
Communication, honesty, and self-awareness are the cornerstone virtues of polyamory. You don't have to be perfect in these areas (who is?) but if you've got a major weak spot in any of them, you probably want to work on it before you try being actively polyamorous.
It's probably also a good idea to take lots of time to explore the territory before you begin another relationship. Read, discuss, talk with your partner about ideas and expectations, what model of polyamory you think each of you would be most comfortable with, and all that stuff.
The fact that your partner allows you to be exactly who you are, and is supportive of your exploring this, is a really good sign. Be patient with yourself and with each other, expect some ups and downs, and try to enjoy the ride. Best of luck to you!