I also own a 1986 Toyota Corolla and I helped my dad remove my carby, clean it out, replace the gaskets and put it back in, so I can tell you what I know.
Step 1: Buy a carby kit (with all the gaskets and other bits and pieces)
Step 2: Remove the air filter completely.
Step 3: Draw a diagram of the carby and the connecting tubes. Number each tube on the diagram as you remove it and put a bit of tape around each tube with the corresponding number so you know which ones go where when returning it to the car.
Step 4: RE: Your fuel line: Dad just sat the tube so the end was vertical (So the fuel wouldn't just spill out all over the ground and over the engine) and tied an old rag over the end - keeping it vertical all the time. You're going to make a bit of mess, you can't really avoid that.
Step 4: Clean out the jets with a high pressure air gun, remove and replace the gaskets, check your float level, (and there are probably a couple of other bits here that Dad did without me there).
Step 5: Put carby back together and using your diagram, have fun connecting the tubes back. Make sure you put them firmly back on, the tube to my EGR valve came loose and caused all sorts of problems!
Step 6: Put the air filter back on and you're done.
I guess that's basically the process, you need more details about pulling the carby apart and physically cleaning it.
Hope this helps a bit, even if it doesn't, at least it shows I know a little more about cars than the average girl