Getting a copy of the original Total Annihilation is much easier than getting a copy of either expansion. It was available in the Commander Pack, which included the original game, both expansions, and a special edition strategy guide. It can be purchased from Amazon's new & used section fairly regularly alone or in the Commander Pack (as of Feb. 2008). Since Battle Tactics was generally less popular than The Core Contingency, there may be fewer copies of Battle Tactics available, but it is also generally cheaper.
The original Total Annihilation needs to be installed before Battle Tactics. If you plan on installing The Core Contingency, it should also be installed before Battle Tactics. The 3.1c patch and any downloaded units and maps should be installed after Battle Tactics (see Total Annihilation/Getting Started for more information.
The primary means of playing Battle Tactics is through the Campaign interface. Select Single from the main menu, New Campaign from the Single Player menu, select a side (ARM or Core), and scroll the Choose Campaign list (the options should include Campaign, Campaign - Core Contingency (if installed), Battle Tactics - Long, Battle Tactics - Medium, Battle Tactics - Short, and Battle Tactics - Very Short.
You can then scroll through the Choose Game list to choose a mission from the campaign you have chosen.
Battle Tactics (and the 3.1c patch for original Total Annihilation) opens up all of the missions to be selected for play. The primary reason for this is that the Battle Tactics missions were intended to be played in any order. While it is recommended that players new to Total Annihilation start with the Very Short missions to learn basic tactics (even before the main Total Annihilation campaign), you are also encouraged to explore the missions available and choose whatever missions may be most helpful to you.
There are a handful of multiplayer/skirmish maps, and the means for playing on those maps are as usual for multiplayer/skirmish games. Start a multiplayer or skirmish game (skirmish games are under the Single Player menu) and select the map, or join a multiplayer game using the map.