1a. Flash and sprue: Straight out of the package, there is a lot of excess material attached to the mini from the mold lines (flash) and injection points (sprue). Use a hobby knife or needle file to take it all off.
1b. If you have a used mini that has been painted, you will want to strip the old paint. The easiest way to do this is to set the mini in a bath of Pine Sol household cleaner overnight. In the morning, the paint should wash right off. It's even easier if you use an old toothbrush to help scrub the mini clean.
1c. Assembly and hex base. Most minis need assembling, so you should have some glue on hand. In my experience, it is best to use epoxy (the strongest method) or Devcon Weldit (easiest method). Superglue or gap-filling CA are ok for small parts, but they do not hold large joints together very well unless you have an accelerator. If you want a hex base on the mini, you should glue it on now.
1d. Antennae. Many minis look better if they have antennae installed (consult the TR pics to see which ones and where). To install antennae that won't fall off before you finish painting the mini, you should drill a hole to seat it in. Using a 1/32" drill bit, make a hole 1/8" to 1/4" deep (space permitting) and glue the antenna into the hole. NOTE: this procedure can be pretty tricky, and is not recommended for beginners.
1e. Prime the mini. Get a can of spray primer, and hose down the mini. Do not put a thick coat on! For this step, it is better too see bare metal underneath (thin coat) than to obscure the detail (thick coat). The primer has 2 purposes: it makes the paint show up a little better, and it makes the paint stick to the metal much better. Proper priming is your number one defense against chipping or flaking paint.
2a. Decide what colors you want your paint job to be, and get the paint. Good brands are Ral Partha, Polly-S, and Testors Model Master Acryl. In general, you will always want to use a water-based paint. Citadel paints (Games Workshop / 40k stuff) works ok sometimes too, but a few of the colors have reacted with my varnish, so I don't recommend it. Do NOT use standard Testors paints (or any enamels), and stay away from the Model Master paints that are not Acryl (a Testors trademark name for their good acrylic paints).
2b. Start with the lighter color. Using a relatively large brush (size 0 works well), paint the entire mini. Do NOT put on a thick coat of paint... several thin coats will look much better and last longer. Make sure this paint is dry before going to the next step.
2c. Add the other colors, in order of light to dark. For most paint jobs, a size 000 paintbrush is sufficient for this. If you have access to an airbrush for camo work, this is where you'd do that (Aztek A430 with Fine or Acrylic nozzle works well).
3a. Inking / wetbrush (optional). Using inks or watered-down paints, you can use a small paintbrush (size 10-0) to put dark paint in the cracks and crevices of the mini. This adds a lot of depth to the figure.
An alternate way to do this is to take some watered down dark gray or black and use a large brush to spread it all over the figure (known as washing). The paint will naturally settle into the cracks. It takes practice to make this technique look great, but it is quick and fairly simple. Generally, I dab the brush lightly in the paint, dip the brush in water, then brush the mixture over the mini. Make sure there is more water than paint, or else you could turn your whole mech black. :)
3b. Drybrush the mini. Make sure all other paint is dry before doing this, Use a large (size 0 or bigger) brush, and make sure it is dry too. Pick a lighter shade of the main color on the mech or white. Dip the brush in this color, then rub the brush around on a paper towel until almost all of the paint is gone. Then, brush the mostly-dry brush over the edges and corners of your mini. Traces of the lighter color will cling to the raised surfaces. This technique is important to practice, and if done well it can make a poorly-painted mini look professional. You can also experiment with drybrushing multiple colors onto the figure, each successive color being lighter and then ending with white.
4a. Paint gun openings and missile tubes. These will usually be black or dark gray, and you can paint them with a size 10-0 brush.
4b. Paint cockpit, and other small areas of detail. Again, a size 10-0 brush work well for this. Depending on how you want to do the cockpit, you might want to drybrush it after this.
4c. The decision point. Here, you judge your mech. If you like how it is coming out, proceed to the next step. If you don't like it, put it in the Pine Sol and return to step 1.
4d. Add decals. You can get some custom decals on ebay or elsewhere without too much expense. Or, you can use decals from small scale (1/72 or smaller) model airplane kits. For mechs, it is usually cool to put some numbers and some sort of unit insignia or marking on each mech. Waterslide type decals are the best by far, so get that kind and follow the instructions that come with them. It's best to make sure they stay on by using some brush-on Decal Set.
Hose it down with some sealer or varnish. I like either Testors Dull Cote or the Citadel Matte Varnish. In any case, dull / flat kinds look better every time. You might think a gloss paintjob would look cool, but it usually messes up the effect of scale and makes the mini look worse overall. Anyway, like always several small coats are better than one heavy one, but here it isn't important since the varnish is clear. This stuff will help hold on the decals and prevent the paint from chipping.
Methods of doing this vary, and you can experiment to decide what you like. Basically, you get bags of green turf or tiny rocks used for model railroad displays. Spread white glue on your hex base, and dump the terrain onto it. Press it down into the glue to make sure it sticks. After it's dry, brush off any excess material.
Show up to battle and make your friends jealous.