On a sidewinder pulses are a better option. Better than the multi-cannons to take the shield down and still okay for physical damage. So decent all round for the small sidey.
Be careful what you take on in a sidewinder. Don't go somewhere to busy with pilots who could shred you in seconds. I hung around Nav Beacons when I was BH in a Sidewinder. If you're picking on larger ships with a bounty, try and go after one's that are already engaged against security vessels. It'll be easier as they won't focus entirely on you and you have extra fire power helping you.
I had a free Eagle so quickly upgraded that, which is great fun to fly. In fact I kept it, I enjoy flying it that much and still do a bit of bounting hunting in it against NPC's. Capable little craft, but has paper thin armour once your shields are down, so be prepared to run when needed. Depending on your earning rate, you could go for a Viper MkIII next which is a decent step up from an Eagle and can pack a punch with the right loadout. In fact, I've seen a particular ViperMkIII guru taking out winged CMDR's in much larger ships. I'm not even close to that level of skill though.
Again, what to aim for after a Viper depends on your earning rate vs your patience. Vulture, Fer-De-Lance, who knows. Depends what you'd like to do as you progress in the game. I want to get a Vulture, but the initial purchase price of around 3-4m cr is eclipsed by the upgrade costs. So you may decide to use a Diamond Back Scout as a stepping stone from a Viper. The DBS has similar punching power to the Viper, but has much better armour. However, it does have slightly less capable shields. So it depends on if you panic when you're shields are down, or whether you want the extra toughness of the armour.
Everybody will have differing opinions, so as you build up your credits you can try different ships and loadouts and see what feels best for you.