Unless you're playing on Point Blank or something, the AI, especially the Easy AI, should take much longer than 10 minutes to get to you. The AI usually takes like 5 to 10 minutes per (neutral) planet it takes. So you really should have a decent amount of time to get started before you have to fight the enemy, if you're playing a 1v1.
In larger games (like Random Large or Random Huge) , there are fewer planets per player to go around, but the galaxy radius doesn't increase that much as well, meaning that there's much less time to prepare. If that's the kind of thing you've been doing I highly recommend starting out attempting 1v1 games instead, on "Random Small" maps.
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If you're having trouble killing the Easy AI, try this. Play as TEC in a 1v1 random small against an Easy Researcher AI, with Pirates off and Quick Start on. Queue an Akkan as your free capital ship, build 2 military labs, research LRM frigates, and set your homeworld's Rally Point onto your Akkan capital ship.
Queue as many LRM frigates as you can. Once you've built as many as your supply can hold, go up a supply level and build more. Repeat.
Your scouts should have scouted a good amount of the galaxy by now. Once it's built, send your Akkan towards the enemy, colonizing planets as you go. Your LRM frigates will be supporting your cap. (Remember to develop Civilian Infrastructure on your newly colonized planets ASAP, or you will lose a lot of credits to underdevelopment) Once you start running into your enemy, you should have 30-70 LRM frigates along with your Akkan. They will completely decimate whatever fleet the AI has and you should be able to steamroll their homeworld in well under an hour, even with no micromanagement at all.
then I built a missile platform
For learning the game, try relying on a fleet, not on static defenses. Don't build turrets except as a last resort.
Small combat ships take 15-25 seconds to build. Even were the AI to attack you 10 minutes in (which it shouldn't), you should be able to have a good number of ships yourself by then.
Also: If you haven't already, TURN PIRATES OFF. They're very difficult for new players to deal with and impede learning of the game.