If you're having trouble in the first 5-10 minutes of the game, you can't tell time.
The AI doesn't even attack for the first 5-10 minutes, even if you set them to Unfair difficulty. It takes that long just to hop three planets over to you, so add that time to how long it'd take to build a reasonable fleet. If that happened, they'd end up sending all their ships at you and someone else could wipe them out easily.
In short: You're doing it wrong.
A lot of people try to expand really fast. *You simply cannot do this. Period.* I'm not saying you have to expand so painfully slow that it takes you two days to take an entire galaxy, but don't try to build three colonizers and wonder why you die before the second autosave.
Build a capital ship, preferably the "Battleship" one (The one clearly meant for fighting, not support. The one that does *not* have the "colonize" ability.) and lead it to some other nearby planets. MAKE SURE YOUR PLANETS ARE GROUPED TIGHTLY TOGETHER. You need to be able to get that capital ship to any planet you own /fast/ in case you do happen to get rushed early on. Ignore how valuable planets are if they're two or three hops away. Trying to take them too fast will get you screwed over. Someone will either attack your home planet while you're away and take it, or they'll just take the new planet you got when you go to defend your home against someone else and it'll end up getting you doubleteamed.
The colonizer frigate's ability recharges at twice the speed you should be colonizing early on. In case that doesn't make things obvious, let me make this clear: You do NOT need the colonizer capital ship as your first capital!
Once you take the 3-4 planets one jump away from your homeworld, then you can start to expand a bit further. Some people like to try to take valuable planets next, others like to expand away from the biggest threat. Whichever you pick, you need to be sure you can actually hold the position. Don't try to take something so your opponent can't get it with the mindset of "Well if I take it by the time they get here it'll have 1,500 HP and take forever for them to bomb it.". That doesn't mean anything. They can still bring 20 frigates over, keep you from clearing them out, and then bomb it as slow as they want.
Finally, don't play TEC Rebels unless you really know what you're doing. The more you're hated by other factions, the more the AI tries to kill you. TEC Rebels start with a -10 with everyone except other TEC Rebels.