As mentioned above, you need to have a sufficient supply of cruor to get started as a key leech. And unfortunately without having the necessary key items to provide auto refresh, you won't be able to be much of a healer there either.
To get started with abyssea, you'll have to do a few quests, mostly involving seeing various cutscenes in Port Jeuno and the different Cavernous Maws. Once you've completed the basic requirements, you'll start accumulating Traverser Stones which provide the time used inside of Abyssea. So even if you can't get started right away in Abyssea, you'll still want to start accumulating Stones for later use. Once you've done the steps necessary, they'll accumulate automatically and you won't have to do anything further except ask the NPC to hand them over to you as needed.
During the time that your accumulating stones, you can solo WHM on Fields of Valor training regiments to get a few levels while seeking for a pt. You may get lucky and be invited to a level Sync party somewhere and get a few levels that way. Hopefully by the time your WHM is in the 50's you'll start being able to be a backup healer for abyssea and may be able to get away with that for a few levels.
Completing certain missions (Windurst story missions and A Shantotto Ascension) will grant you key items called Atma that you can use in Abyssea to grant you a 5 mp/tic auto refresh that stacks with other forms of refresh.
Edit: Make sure you have the WoTG expansion, and have access to the East Ronfaure [S] zone for level sync purposes.
This post has been edited by Nanako: 24 November 2010 - 12:56 PM