First, download the Technic launcher:
http://www.technicpack.net/downloadIt's a launcher for a collection of modpacks (not just tekkit) that you can play at your leisure. Some of them are pretty fun. Anyway, after you log in you need to scroll down the list on the left-hand side until you see tekkit classic. Click the "Play" button on the right-hand side and it'll begin automatically downloading all the modpacks you need to play the game. Once that's done, all you have to do is wait for the Minecraft window to pop up, hit multiplayer, and connect to the server (the IP is the same as the vanilla world, port is 25568. So what you should connect to will look like ***.***.***.***:25568).
Now, tekkit's a pretty big modpack, and there's a lot of stuff you need to know about it to play it well. I'm going to give a brief introduction to the main two mods of tekkit.
First up is Buildcraft. This mod adds machinery to Minecraft. The first machines you'll want to get are a macerator, that turns any ore you collect into two dust of that ore (1 dust = 1 ingot when smelted), doubling the amount of resources you collect if you use a macerator. Next is a coal generator, to power your macerator. You'll also need to find a rubber tree and collect sap from it to create rubber to insulate copper wiring to link up your coal generator and macerator.
You may be wondering why I'm not spelling out any recipes for these things. That would be because tekkit comes with the NotEnoughItems mod, which adds a searchable list of all in-game items (including all the mod-specific items) to your inventory. You can also view the crafting recipe for any item (assuming it can be crafted) by pressing R on the the item (or potentially just by clicking if you're not an op).
Next up is Equivalent Exchange. This is a really powerful mod, and if you know what you're doing with it you can essentially enter creative mode with it in no time at all. EE adds a value to every EE and vanilla item in Minecraft, and allows you to transmute items based on this value. So, for example, if diamonds have a value of 8192 (which they do, I have it memorized), and cobblestone has a value of 1, you can make 8192 cobblestone from 1 diamond or 1 diamond from 8192 cobblestone. It also adds very powerful armor and weapons, as well as magical rings (one of which lets you fly).
The three things you're going to want to make ASAP with this mod are an energy collector, an energy condenser, and a transmutation tablet. Energy collectors collect energy from light (even from torches), and if they're placed next to an energy condenser will transfer their energy to it. In an energy condenser, you can place any item that has an EE value, and the energy collected from collectors will build up until that item is created for free! This can take time, and there are many tiers of collectors, but it's extremely handy in mid/late game. Transmutation tablets are important because they have memory; if you put an item in a transmutation tablet, it "knows" that item, and allows you to create it if you put in enough items that add up to the first item's EE value. This is useful if you accidentally convert all your iron into diamonds using an energy condenser, for example, which will probably happen more often than you'd like.
That's about as brief of an introduction as I can give. If you have any questions, feel free to ask and I'll do my best to point you in the right direction.
EDIT: Also, a bit of info about rubber trees. You don't want to mess this up.
Here's what they look like:

Notice the unique stack of leaves on the very top of the tree.
DON'T BREAK THESE. They're necessary for the tree to produce sticky resin. You can break all other leaves except those to get saplings.
Here's what sticky resin looks like on a rubber tree:

It's that yellow part on the trunk. Right click on that with a tree tap and you'll get sticky resin.