health vials are instant use, they'll be applied automatically.
the amount of spaces in your backpack is shown when you press I, there should be a x/y number in the top right showing you how full you are.
You can accept as many missions as are available but you can only have one active at a time, make sure you visit bounty boards and talk to mission givers regularly as they get new missions frequently.
Artifacts are instant use, click them and they'll apply. If they aren't high enough level then you'll get a message saying you can't use it so you'll be better off selling it as it's too low for you to use.
Class mods, grenade mods and shields show up to the right of your equipped weapons in your inventory, compare them with new ones frequently to see which is better, for my main class mod i use one that benefits my team more (as we play multiplayer more than anything else) if you are playing alone you might be better off with one that benefits you more than a potential team.