Make your forces into two groups. One for the zealots and dragoons, the other for the templar. Move everyone to the northwest, right into the zerg creep. Destroy the zerglings then focus everyone on the sunken colony. Now head on west. When you reach the ramp in the screenshot, you will have to fight some zerglings on top of it, but if you ascend you'll be able to avoid the sunken colony. Do this!

Proceed further northwest and you'll come to two ramps. Take the left one with all your fighting units. Let them destroy the zerglings. Don't worry if you take many losses because it won't matter soon. Now go just a little north toward the area in the screenshot. Use the templar to go to use the psionic storm on the ground at the foot of the ramp. You should hear some satisfying squeals as the burrowed zerglings die. Even if you only kill a couple you weakened them. The rest of your units should be able to finish them. Note that you do have enough energy with this templar to cast that spell twice.

Now simply go down the ramp and curve around to the northwest. You'll be on the platform where the Protoss base is located. After some dialog, you'll be given control of the base. Build up ten probes total on minerals and get three on gas. You can build powerful archons now, and they will serve you well for getting Raynor and Tassadar to the beacon. Follow the tech tree to get to the templar archives. While that's warping in you should get another gateway to speed up unit production. Along the way build some dragoons and photon cannons (to the northeast as in the picture) so they can defend your base from the constantly attacking zerg.

It takes two templars to become an archon, so you're going to want about eight of them. Note you started with two, so you're a quarter of the way there. Four archons will really mow through the zerg, and with about ten zealots (you will be waiting on gas, so building these is no chore) to back them up they won't have any trouble making it to the beacon. I recommend you keep Raynor and Tassadar well behind your forces for your journey to the beacon because neither is very powerful. Getting the plasma shields upgrade to level two for your units would be a very wise thing to do. Especially since the zerg along the way are set and aren't going to be increasing in number. It's worth the wait. One useful thing you can do along the way is use the psionic storm of Tassadar to soften enemies. I'll let you know of the great places to use it.

Now you'll see it's not as simple as following the path back to where you started. You will take that path, but it's now infested with many more zerg than before. Head down the ramp west of your base. Make sure it's the one you came up to get to your destination earlier. If you take Tassadar just to the bottom left-hand corner of the ramp you can see some hydralisks sitting on the platform above, in your path. This is a good spot to use the psionic storm. Try to center it on them, so it causes as much damage as possible. If you inch him further southeast, you'll see another group of hydralisks. Stop when you see just one of them then use the storm just behind it. You'll smash them all (in theory) and be able to proceed still unchallenged. Now go up the ramp.

Put your archons in the lead and kill the queen, then wait for everyone to catch up. When you get to the spot with the two ramps again, this time make Raynor use the spider mines just at the top of the ramp (all three!) make sure there's nothing that will keep him from escaping, then head southeast and get the zerg to follow you. They'll at least be damaged, and you can then use your units to finish them off. Try to make the zealots take most of the damage. Now you're almost there, just a few more minor trials.

When you reach the area of the sunken colony you bypassed earlier, do the same thing again. Lead with your archons while everyone else waits for them to clear the top of the ramp. Once everyone's up on that platform, proceed back down it with your archons, clearing a path once more. Now your poor archons will once again wade into battle all alone. There are about six mutalisks just around the corner, where the first sunken colony was. Lure them back to the northwest, destroy them, then back off!

Now you can use Tassadar's psionic storm one final time, to great effect. When you started fighting the mutalisks you probably saw some hydralisks right in your road before the beacon. You can inch forward with Tassadar like you did earlier and cast two psionic storms, side by side (to cover a greater area). You should at least hurt them. Now you can retreat tassadar and charge in with your zealots. They can easily finish off the hydralisks. If not, they'll soften them a lot, leaving you able to complete the job with the archons. Now simply destroy the mutalisks and send Raynor and Tassadar to the beacon.