It's great fun!
Personally, I enjoyed the gearing process the most, it's not difficult to obtain BiS (server dependent for the AGM/LFH) for gear and enchants, it doesn't cost thousands but expect it to cost you several hundred gold if you want to be a good level 1 twink.
Your professions are quite entertaining to level up if you have more than 1 account (to duel-box through those 'impossible' zones), but I would imagine it to be a pain if you had to rely on friends/randoms to provide any form of gathering escort service (looks at herbbing in Stonetalon -_- ). You should choose Mining (for stam) and herb (for heals (haste/heal in cata)).
Once fully-geared/enchanted and bags-packed with all possible self-buffs, your ready to do several things:
* PVP in your faction's 'noob' area (Goldshire & Xroads usually). The reaction you get from beating a level 20+ is a thrill (seen a level 32 keyboard turner n00b beaten, my record is a 24 druid), and knowing you can slay almost any lowbie with comparitive ease gives you a warm feeling inside, with or without heirlooms
Make a video of your duels and edit your own masterpiece for Youtube.
* Do some PVE, either solo or with other L1 twinks. Hogger can be solo'd by any fully-geared/buffed L1 twink, providing no add's are attracted (or get any high level for an escort), but 666 HP is no probs for my L1 Rogue. There are many, many Level 11+ elites that PVE groups can target, just requires a little research using filters on wowhead.com to find good 'bosses'. Cata is however removing all possibility of having a healing L1 (I think Chubbs has confirmed this on BETA) so this brings a fresh challenge of bandage-speccing to L1 twinking.
* Go achievement farming. The finest L1 twinks on this forum have over 2,000 achi points, which is more than some people's main's. You either love them or hate them, but they are good boredom-breakers which I guess is the whole idea of twinking in the first place.
* Do all of the above. After a few weeks of (maxxed out) level 1 twinking, things tend to go stale, in particularly if their are no/minimal other L1 twinks on your server, there are only so many level 18 paladin's that you can leave gobsmacked, so you do need something to keep you coming back to your young twink.
Personally, I love the L1 PVP, and really wanted to get started in some PVE but the lack of L1 twinks on my realm proved to be too restrictive on ever starting a PVE group, so solo'ing Hogger became a regular boredom-breaker. Achievement points tend to be easily obtained if you have another account with any high level (pref 80 obv), and even easier if you have a 2-seated flying mount
Relying on friends/randoms to provide escorts is time-consuming, can be costly in gold, and is a drain on the entertainment you are getting from twinking at the lowest-possible level, so ultimately it is just your own imagination/desire that keeps you coming back to your L1.