I really loved flying an Asp and still have a dedicated asp kitted out for exploration, at which, if given the correct load out, it excels at which it should. It's also incredibly good at the other roles as well - however you need to cchange the loadout each time to ensure you're getting the best performance out of it. It can shift a lot of cargo whilst still being able to pack quite a punch if you're interdicted or fancy a quick change of pace.
The main thing the Vulture (seems to be the BMW of the ED universe) has over the Asp are the large weapons which are far more effective at dealing with larger ships, there's no getting away from that. However you can fully A-grade an Asp and it's truly multi-functional. With the six hard points you can use them all at once or if you're using a mix of energy weapons and kinetics you can run with the two energy weapons and two kinetics and keep the other two powered down until you run out of ammunition and then switch to the second two, you can stay in a decent RES for far longer that way and I used to earn a lot more credits per hour that way.
My advice (once you've got sufficient credits) would be to have two shield boosters, an A graded power plant in an Asp can provide sufficient power, just about, to run them and everything else and that can give you a real edge.
Also whilst I'm thinking about it the powerplant and distributor should be your first priority, but upgrade your shields as soon as you can, the basic shields seem to be made out the equivalent of tissue paper and powered off one triple A battery.
Also it sounds awesome