"In all endeavors, you should start at the beginning, proceed until you reach the end, then stop."
Okay. You just discovered Battletech at your FLGS (Friendly Local Game Store) and you want to do everything at the same time, but you don't know what to do first. Hence the quote above.
Start at the beginning. Ask questions of the players. Next, come home and since you're already on this website (DUH!), read everything else here too.
Starting out, I suggest you play with OPM's (Other Peoples' Miniatures) first. Make sure you like the game before you start dropping serious coin on this. If you do decide to start dropping coin, figure out a budget of what you can afford without getting your spouse/significant other upset at you and making you sleep on the couch. If you're single, try to avoid the "air and ramen" diet as well..
The first thing to get in just about every instance I can think of is one of the latest Box Sets. The Beginner Box is just enough to taste Battletech with two miniatures you can paint. Support your FLGS and buy them locally, the link is so you know what you're looking for. The full box set is called "A Game of Armored Combat" (AGOAC) and has twelve miniatures. Both have the basic rules of the game, a history of the BattleTech Universe and what you need to start out playing.
Get the Beginner Box and play a few simple small games at home with a friend or spouse (if they're into wargaming as well). Get used to the mechanics and how things work in the game, and why they do so. Once you have that under your belt, move to the next level. Play with your newfound friends. If they loan you miniatures to play with, treat them as they were your firstborn. You'll understand why later.
Next, buy the book Total Warfare. It has all of the rules related to the game itself. Continue playing with your same Mechs, moving up to some lance vs. lance battles.
Now that you have some experience under your belt, you'll want to expand. You find your stable of Mechs to be limiting and you're eyeing all those blisters on the wall of your FLGS. Don't go buying those quite yet. The nice thing about Battletech is you can proxy Mechs. You can take that Jenner, call it an Atlas and keep on moving. As long as you have the sheet to the Mech you are playing, it doesn't really matter what it is represented by, as long as everybody knows what it really is.
Your next step is to start looking through the fiction of Battletech and discover what era do you want to be in and who your Mechs are fighting for. Do you wish to fight for one of the Great Houses, a Clan, a faction such as Comstar, or do wish to be a mercenary? If you fight for a Great House or Clan, you will be limited as to what Mechs you can use. It is acceptable that the 1st Davion Guards in the post-3057 Fedederated Suns might have a couple of Steiner Mechs, but not a lance or two of Kurita units. That wouldn't "fit" right. If you run as a mercenary, becuase most of your 'Mechs would be salvaged from battles, while others would be purchased from whatever employer you had at the time.
Go through the archives and see what House, Clan, faction or Merc unit tickles your fancy. Pick one by its history, or just because the unit has a cool looking paint job. If you don't see anything you like, do not despair! Make your own unit! Pick an era, and write the history down. No one is stopping you but yourself. Sarna and Camo Specs Online are where you want to start. Sarna is a fan wiki, compiling most of the Battletech Universe into one spot. CamoSpecs Online has examples of Mechs with various units paint schemes.
Once you have determined what your pool of available units are, you will need those record sheets. There are three ways you can obtain record sheets. You can purchase the appropriate collections of record sheets you need, or one of the Mech building computer programs, MegaMek or Solaris Skunk Werks.
You can buy compilations of Mech sheets directly from DriveThruRPG, but you have to buy them in their packages. Each collection offers to all of the factions a little bit. So, you will have to get several volumes to obtain all of your Mechs record sheets. DriveThruRPG sells PDF's, so you can print out the Mechs you need directly without lugging the book to the copy machine every time.
Both Solaris Skunk Werks and MegaMek run under Java, so they can run under Windows, Macintosh and even the Ubuntu distribution of Linux. Both of these are free, but they always appreciate donations. SSW has a vast library of Mechs, but the last update was September 2014. The good news is if it doesn't have a weapon or system, if you know what you're doing, you can add it. MegaMek also allows you to play on the computer, either against AI or another online player. The nice thing about these programs are that if you want to make your own version of a particular Mech, you can modify a design with just a few clicks and print out copies of whatever Mech you want or have.
Now you can start to expand your horizons. Use the record sheets to find out what Mechs you like and fit your play style. Once you can write down 36-40 Inner Sphere Mechs, or about the same number of Clan Mechs (depending on your ultimate unit configuration), NOW you can start buying those shiny Mechs on the wall.
In all of the miniatures games I play, my primary advice to a beginner is "build your list, and then buy those units ONLY." If you do the "Ooh, shiny!" and buy units that look cool (like my son) you will end up with a lot of units you will never use.
I recommend a battalion sized pool of Mechs for the simple reason that you will have a wide variety of Mechs to draw from when putting together a force for a scenario. Light, medium, heavy and assault Mechs all have different advantages and disadvantages. Each Mech will do something well and other things poorly. What you use may also depend on the style of scenario you will play. At this point, the sky is the limit.
You now have a battalion of Mechs (at least on paper), a history to go with them and a paint scheme. By the way, painted Mechs fight better and look cooler. I cover everything necessary about building and painting your miniatures in my e-book, See Mark Paint.
Good luck!