I've been trying out the chars in BL 1 and am sticking to Lilith. Seeing as you can be more proficient at certain weapons, I'm wondering - can you spread yourself too thin if you swap between a large variety of weapons?
Annoyingly enough, you can quite easily gain "general" XP (which raises your level) much faster than your individual weapon proficiencies. If you evenly switch between weapons you might not notice a huge difference, but what WILL be noticeable is if you neglect a certain weapon type and suddenly try to use it. A weapon with significantly lower proficiency will perform noticeably worse, which can cause a vicious cycle of you avoiding using that weapon, which means it won't gain proficiency.
If you really enjoy using all the weapons I'm not sure I would recommend deliberately not using some weapons in order to boost specific ones because of that effect; if you focus on weapons, they WILL grow faster and it will be noticeable, but you will make those other weapon types MUCH harder to use later on.
If you plan to play though playthrough 2 and all DLC, I actually recommend that you spread yourself thin to keep all weapon proficiency nearly even, because once you cap out a weapon (50 is the max level for weapons, even though the player level cap is 69 with the General Knoxx DLC) all weapon XP earned while using that gun goes to complete waste.
tl;dr the game will be a bit easier midgame if you focus on just a few weapons, but if you plan to play playthrough 2 and all DLCs and hunt for loot, you'll find you have much more flexibility if you spread around weapon experience.