Congratulations. The first victory is a big step in this game.
For the Overlord phase 2, it also matter quite a bit what you bring. I love to hack Ablative Gatling ships in my fleet because they are so useful, and have a x5 damage bonus against phase 2. Bringing five hundreds Mark six or seven really makes a difference. It used to be worse when they had x11 multiplier.
On a less obvious note: the battlestation. I always pair the starting one with my support fleet nowadays. Upgrade it on the fleet screen once (boosting its turrets and increasing its DPS) then once on the tech screen, you'll get something able to clear most early-game threats even before bringing turrets in.
Then at the Endgame you can bring turrets backup against the Overlord. The Electic Battlestation with Nucleonic is a favorite of mine, because a full Mark 7 cap of Nucleonic is bad new for the Overlord. Very useful if phase 2 ends up running away too. I quick-conquer worlds Phase 2 run to (in the rare cases it manages to flee), build a station to remove the 'Building times for mark >3 on hostile worlds' and drop Nucleonic in range of the Overlord. Very efficient.
Another relevant trick is to beachhead (ie, build turrets with battlestations) on an AI world(s) a couple hops away from the AI Homeworlds. Being AI-held, the Warden fleet can show up, so when the AI Homeworlds has lot of units it's a good way to whittle it down or give you an opening. I have had particularly great success with this trick when the AI homeworld has a single wormhole.
