How to make the big switch?

I had the same thing, or at least similar looking stuff, break out in my tank along with some hair algae in the early months after I set it up. I bought a big bunch of grape caulerpa for my tang and within a week the nuisance stuff was gone. The caulerpa devoured all the nutrients the nuisance algae was utilizing and it wasn't long before it was completely gone. I have since then kept some form of macro algae in my system and have found that cheato is effective in controling/preventing a nuisance outbreak but doesn't grow fast enough to eliminate an already existing problem. Caulerpa grows much quicker and more aggressively and consequently does a much better job robbing the nuisance algae of any available nutrients - cheato grows too slowly to keep up with the nusiance algae.

I have a 55g and, at the time, didn't have much water circulation and only a simple 4-stage canister filter with a bright light (needless to say nuisance algae heaven). I swear that the caulerpa was the only thing that saved my tank from total takeover. I don't know if you can get it in your area but if you can, get a whole bunch of caulerpa and just throw it in your tank.

 

My xenias I just cut back attached to the new rock in just a day or so. I was surprised how quick it was. I didn't use rubber bands I just found an area of little flow in my fuge and used some small chunks of rock to hold it in please where I made my cut and the next day they were attached to the rocks.

 

Kylie most likely your new tank will go into a cycle or two after you put your live rock and corals in it!
Use a turkey baster and blast the stuff off your rocks and corals, then suck the free floating algae up with it and remove it from the tank.
Do your normal water changes after you blast your rocks and corals!
I would be patient, don't let it over run your corals but let it do its thing. Give it time and it will disappear on its own. Yes it will look ugly for a month or so but you wont put any undue stress on your tank!
That is what I did for My 120g when I had my out break.