What is the best tactics for dealing with Slylandro Probes at the start?

Spathi Eluder FTW - unequivocally.

Make the probe chase you. Wait until it almost closes the distance and begins its "up-down-up-down" "impatient" dance as I call it. When that happens, use the Backwards Utilized Tracking Torpedoes (BUTTs) and fire off all three, and as many more as you can until the probe breaks away due to damage or because you've gotten too far away from it.

As soon as the probe starts to go away from you (to try to loop around the map and get you from the other direction) turn around and chase it until it either closes with you on your new back side, or it turns around to give chase again, at which point you turn around and let it chase you until it starts doing its "up-down-up-down" thing again.

Continue until probe is destroyed, without suffering any damage. If you use the slow Earthling Cruiser, your missiles could miss and the probe could hit you with its matter reclamator.

The Earth cruiser, despite otherwise sucking pretty bad, is actually decent against Slylandro probes. Practice in quick battle mode for a while, just a lot of Earth cruisers vs. Slylandro probes, until you're good at it. The strategy I use against the probes:

• Keep your distance. Basically stay as far away as the battle map lets you. The probes have no long range weapons, just the lightning thing that only works if the probe is right on top of you. Also, your missiles work better if they have a long way to travel (unintuitive but true!). 3 missile hits will kill a probe.
• If it closes and blasts you a bit with the lightning, try to keep it away from asteroids. The lightning has a limited charge; a probe can't kill an Earth cruiser on one charge. It will recharge if it hits an asteroid, but it generally isn't smart enough to go get it. So once it's blown its wad, it sometimes just buzzes back and forth next to you without doing anything... as long as you stay away from the asteroids. Feel free to blast it with the point defense laser when it closes in, too.
• The point defense laser can also be used to blow up asteroids, if you somehow beat the probe to them. Not sure how I recommend doing this.
• Supposedly the probe will back off if you aim your guns at it; this allows the Spathi ship to be extremely effective with the BUTT missiles. I didn't know this but this source claims it is so.
• Go visit the Slylandro planet as soon as you can manage it. They live in the north-west corner of the galaxy, on a gas giant planet. Prioritize fuel and speed upgrades until you can get there, and they'll give you the code to make the probes self-destruct. Alternatively, one of the exits from Quasispace is pretty close to the Slylandro planet, but you'll want to get the Portal Spawner first so that you can use Quasispace to get home as well. As an added bonus, the probes are the fastest NPCs in hyper (as fast as the Progenitor ship with 10/11 speed upgrades), so once you're fast enough to dodge them, you can run away from anything else in space.

Befriend the Pkunk as quickly as you can, the ZoqFotPik as soon as you get their distress call, and the Orz next. All these ships are capable of taking down the probes in the early game. Only use your flagship against one (in the early game) if the probe has only one or two crew left.

Among the early-game ships, I favor the Pkunk against probes, because they are fast enough to force the probes to waste lightning without hitting you, and you can actually outrun them. Also you can ram them and unload a volley. Two volleys can kill a probe, one if you're lucky. Also, once the probe exhausts battery it will go find the nearest asteroid. If it gets you down to half crew, use its refuel run as an opportunity to get some distance between you and escape that fighter. Pkunk is the only low-crew ship that can feasibly escape from a probe.

Using the Orz, rotate the main gun to your 6 o'clock and fire as you run away from it. The probe will not come straight on when you do this, reducing the speed differential a bit. Send out vac-suit attackers also to disrupt its attack pattern - no more than two at a time. Use the gravity well to your advantage also; with vac-suits in play it may actually ram the planet a couple of times.

Using ZoqFotPik, (as in any situation where your ship is slower than the enemy,) stay close to the gravity well. Ram it at the correct time and use the stinger attack. If you hit it with the stinger, it's one and done, but it takes skill and a little luck. Also, since the probe will run away when you point your gun at it, with practice you can use this fact to "pin" it between your gun and the planet.

However, there is no need to fight the probes at all until you're ready for them. With full thrusters (easy to get early) you can outrun them in Hyperspace. It's easy to tell which ones they are from their speed, and you can always keep enough distance that they can't catch you even when you visit a star system. Once you have upgraded main guns and at least one homing device on your flagship, probes are no longer a threat; kill 'em all!