There is no specific way to level up fast. (Well, there were exploits like that RSVP mission, but they are grinding and not fun.)
You just need to take missions slower - i.e. check level on main missions - if the main mission is of higher level than you, go look for some side missions. Generally, if you do all of the side missions, you should have plenty of extra experience and be at least of equal level to main quests.
Shield points do not depend on your level - only on the shield equipped. To get high-level shields and other equipment, you need to either farm an area with monsters of decent level, or frequent vending machines in such areas. For example, if you are on a Dam Fine Rescue quest and level 12, there is no point to hang out in Three Horns Divide - the area, enemies and vending machines are all level 8, and you'll never get a good drop there.
- To get more HP, level up by doing more side quests.
- To get stronger shields, fight stronger enemies, ideally of your own level.