During the quest to recruit the Professor, usually one of the first quests players do, the Professor will give you a new Pistol. At that point you will have two Pistols to choose from. You can then go to the weapon locker in the Professor's clinic and change your own weapon or that of your squad between the Carnifex (the new weapon the Professor gave you) and the Predator (your starter Pistol). For Shepard I would recommend using the Predator. Even though it does less damage per shot, it has a higher RoF than the Carnifex, it has a clip that is twice as large, 12 vs 6, and it has twice as many spare clips, meaning that one load-out will do more damage than the Carnifex. For the squad, since they never run out of ammo, I would recommend equipping the Carnifex.
Another mission where you will pick up a new weapon is the Archangel mission. There you will pick up the Vindicator assault rifle which is much better than the Avenger.
If you have the Stolen Memory DLC, you get a new squad mate, Kasumi, and if you do her mission you will get a new SMG, the Locust. The Locust is much better on Shepard than the starting Shuriken. Later on, you can pick up another SMG called the Tempest. It is a good weapon, but is very inaccurate for long range shooting.
There is at least one DLC pack that provides alternate weapons including the Mattock assault rifle (very good for a Soldier) the Geth Plasma Shotgun (a longer range shotgun that is good on squad mates) and the Phalanx pistol.
If you do the mission on Haestrom on hardcore or insanity difficulty you can get the Geth Pulse Rifle at the end of the mission, although it is not really a good weapon.