Because I'm allied with so many different pirates, is there any way to pit them against each other?
If you know where their stations are, you can try posting attack missions targeting them. Sometimes attack mission postings will show up in the advisor suggestions, but otherwise I think you'll need to select the target station from the main window and click a button at the bottom of the selection details panel in the lower left corner of the screen to post the mission.
What would be good start for a push against the pirates in terms of fleet size/ship size?
I personally start going after pirate spaceports once I have about 10 size ~300 ships in a single fleet; having additional ships in the space navy or participating in the attack can be useful, but is not in my opinion necessary unless you run into a big mob of pirate ships.
When attacking pirate spaceports, you might want to consider jumping your fleet into the system some way away from the station you want to attack just in case there are some pirate warships in the area, as doing so is likely to draw at least some of the pirate ships away from the station if they are present and combat ready. Missile-heavy and torpedo-heavy armaments tend to allow your ships to sit out of range of most of the station's armament, but be aware that blaster-armed pirate ships can often maul such ships if the pirate ships catch them.
Sample early-game system defense ship:
5 standard armor, 5 corvidian shields, 5 pulse blasters, 2 basic space reactors, 7 ion thrusters, 1 directional thruster, 5 small fuel cells, 1 warp bubble generator (gerax greatly preferred; this setup does not have enough excess reactor output to reach full speed with warp bubbles but does have enough for an unupgraded gerax), 1 command center, 4 life support, 4 hab. This is a size ~210 ship which costs somewhere around 750 credits/year and 3900 credits to buy if most strategic resources cost ~0.8 credits per unit, but you'll probably pay a bit more in the early game due to higher resource costs. A squadron of 5 of those will probably do a decent job fending off pirate raiders, unless three or four come at once. You'll need construction size 230 to build this, and it's perhaps a bit on the slow side so I might consider filling some of the leftover space with more drives, though be aware you're not going to have the power to run at full sprint and fire weapons at maximum rate if you do so. I'd also consider fitting an energy collector to conserve fuel when not in combat.
If you want a ship that'll do a decent job against pirate spaceports, take the sample system defense ship, replace the pulse blasters with 5-10 seeker or concussion missiles, set the design's engagement preferences to standoff/standoff, add whatever else you need to add to make the design legal, and maybe make it a bit faster sublight. It'll be a bit larger and a bit more expensive, and it won't do very well if it gets caught by blaster-armed pirate ships, but a squadron of 10 or so such ships can take down pirate spaceports relatively safely, especially if the pirates have torpedoes on the station instead of missiles.
As for running with no pirate protection at all from the very start, that is not exactly the easiest way to play the game :)
On the other hand, for 9k per annum, jamesbaxter can probably operate a squadron of 10 or 15 ships actually capable of fighting the pirates, rather than subsidizing a pirate fleet of 10 or 15 ships which will almost inevitably be used against him at some point in the future.
< Message edited by Aeson -- 4/17/2017 9:34:41 PM >