Hints & Tips

Join a good alliance.
Just being in a big, strong alliance isn't the answer for everyone. Don't be afraid to jump around until you find one that has good chatter in alliance chat, where people get along with and support each other. Try to find one that fits your play style. Some focus on PvP, events, early level players, etc.
Don't level too fast!
There are good times to spend your speed ups, such as to get points for events and such as that. You have to be careful though about getting stuck in a loop where you don't have resources because you advanced so fast, so you have to spend money for resources, and basically end up in a rut where you have to either spend money or wait long periods of time in order to do anything. You will find that if you pace yourself, you may be a little weaker than you would otherwise, but the game progresses more naturally. Another thing to note is that there are certain milestone levels. Swarm ship daily goals will go up by two levels on each even level, and they get hard fast. Also, your daily hostile objectives go up every level.

NOTE: This is different than the daily faction objectives for both hostiles and mining, those are laid out here.

Level 13
Up til now, you have the option to transfer to a different server. Once you reach this level you must stay on the current server.
Level 15
This is where PvP raiding of bases comes into play, you are making a huge mistake if you upgrade to operations level 15 before ALL of your buildings and researches are as high as you can get them. You will get a 7 day free peace shield when you hit operations level 15. It is best if you can use that time to upgrade all the buildings and researches as much as possible for level 15, so you'll be a strong level 15 instead of a weak one.
Level 21
Up til now, the PvP level ranges are pretty tight. You can check the ranges at pvptarget.com, but I'll warn you that once you hit operations level 21, you will be vulnerable to people who are 10x as strong as you.
Level 23
Armada Control Center
Level 25
Additional Research and Refining related to Territories and Alliance Starbases
Level 26
Tier 1 faction warships
PvE loot protection
Protected Cargo is very import to keep an eye on with your ships, whether you are mining or grinding. That being said, you only lose the amount that's over the protected limit if you are killed in PvP. After quite a bit of testing, I can confidently say that if you are killed by PvE enemies, you keep all your loot regardless.
Rogue faction - Stella unlock
Don't buy this ship for cash. It will seem tempting, but you're shooting yourself in the foot. You really need to save up 210 Rogue Credits to unlock the Outlaws Part 2 mission in the Rogue Faction Store. Completing this mission will get you the blueprints. If you want a second one, and want to shell out the cash for it, then by all means do so, as it will speed up progression. That being said, you can use the stella to more easily get more credits and unlock a second. There is a whole level up / scrap mechanic on this ship, but I'm not going into those details here.
Armadas
As much as possible, try to save your armada tokens until you can do 4 person armadas at a minimum. This will let you get access to slightly harder targets, since you have more people, but it also lets more people have a chance to participate, especially during events that need armada grinding for alliance goals. Chances are you will not be the highest level player, so you can slip in on bigger armadas at first, and run your own once you can hold 4 people. Another thing to note is that the higher level your Armada Control Center (ACC) the more of a damage bonus you get against armada targets, so that's an added bonus.