1 - There are two sorts of energy - Mist (ME) and Crystal (CE). Mist energy is represented by the smaller number in white at the bottom right of the UI; it refills to a max of 100 over the course of 22hours (24, depending on your source). You can obtain ME 'tanks' or refills (100) for completing certain objectives, such as gaining access to T2 and T3; your very first mist tank refill is for reaching the T1 terminal.

The way to get over 100 energy is via Crystal energy. This is the larger set of numbers in blue in your UI. There are a few ways to get CE - pay for it with real money/buy it with crowns/trade with other players. This form of energy doesn't renew itself like ME, but you can hold onto it indefinitely and in a larger quantity than the 100 ME. This is necessary for crafting higher levels of gear, due to crafting costs going over 100 energy.

2 - You don't level up; rather, your gear does. You gain heat from killing monsters or bashing open treasure chests, and this heat is what levels your gear from 1-10. As you level your gear's heat, you will gain small increases in the stats as well as some bonuses (at lv5 and 10, afaik). Additionally, gear comes in various tiers ranging from * (1 star) to ***** (5 star). Most gear can be upgraded to a higher star level via alchemy.

3 - There aren't any quests, per se. You have objectives right from the start - reach the rescue camp; pass the final test and reach Haven; reach the first terminal in Tier 1 of the Clockworks; reach T2; Reach T3. You'll be notified of the requirements (for the last two I mentioned, you may need to talk to the Spiral Warden in the Clockworks for more detail on how you personally can complete the objective).

For more general info, have a look around these forums - a lot of the questions you may have are most likely already here, and therefore likely to have been answered to a degree which should help you. Otherwise, ask any specific questions and no doubt one of the helpful folk that lurk the forums will give you a response.