I find the best weapons at the start of a new character are elemental pistols and sniper rifles. A couple of early quests rewards are pretty nice (for when you get them) for Sniper Rifles, and they’re very effective while you’re leveling in those early areas. For example, the Hammerlock reward for the Bad Hair Day side quest lasted me until around level 12, when I got a different rifle.
A Fire pistol (look for the Maliwan ones called Aegis), while a little ammo hungry, can be useful for the larger bullymongs you have to deal with early on, and I usually end up with one just from world drops. They’re useful.
As @VaultHunter101 says, I wouldn’t limit yourself to just assault rifles, you have very limited ammo for every weapon in early game, it’s sensible to always have one of each weapon type in your backpack for emergencies. If you can find a green rarity Torgue assault rifle, you will find that useful, especially for your first encounters with shielded Nomads.
Hopefully you’ll enjoy the game, I know we all do