There are 3 basic ways to build TM experience and hence TM levels.
First you can complete TM only quests. I don't have a comprehensive guide to them but in general daily quests, card quests and monster quests from assistant hunter and/or love hunter are TM only quests. The side effect, however is that you usually have to hunt monsters for these quests (but not always) which builds regular XP as well.
Next you can fight monsters of much lower level than yourself. They will award more TM XP relative to regular XP so you will advance in TM levels faster than regular XP levels. This is not necessarily the fastest way to close the gap between TM levels and regular levels because your regular level will also slowly advance.
Finally you can do what in my opinion is the best/fastest way to raise TM levels. You can drill. Drill, drill and drill some more. Everytime you dig up an object, any object you receive TM XP. Go to a map where nobody usually does any drilling, like Path to Desert Beach and drill up 5 galder coupons, empty potion bottles and other "useless" junk. If done properly you will find yourself raising a TM level every 15 to 20 minutes or so. The side benefit is of course that you will also make money.
