Best way to start is to start with a tree that will do well indoors under a grow light. You don't want to start with an evergreen of any sort because they need a dormant period in the winter and are a bit harder to keep alive when you first get into it. Believe it or not the best one to start with is to go to Walmart and in the garden section they carry two types of bonsai trees at most all Walmart stores. One is a ginseng ficus (not a fan of those) and the other is a Fukien Tea tree. Get the Fukien Tea tree. It is very hardy and will do very well inside, plus it only costs $10.88.
Here's the thing though. Walmart sells them in plastic pots with potting soil and little pebbles glued all over the top of the soil. If you leave it in that it wont last more than a month or two at best. Bonsai trees need a VERY quick draining soil that is made up of quick drying properties such as a mix of crushed lava rock, sand and even kitty litter. Also the plastic pot is not good for them, they need an actual bonsai pot that is shallow, has drainage holes in it, plus has holes for wiring the tree into the pot.
My suggestion for getting started is before you buy the tree, get a bonsai pot and some bonsai soil. After that then I can help you with what you need to do next such as raking the roots out and trimming and repoting and pruming/wiring etc etc.
This is a good bonsai pot right here for a good price. It has a lot going on for it such as being unglazed, has two drainage holes as well as multiple wiring holes http://www.ebay.com/itm/Oval-Yixing-Zisha-Bonsai-Pot-11-x8-5-x2-/141038592206?pt=Planters_Pots&hash=item20d68e20ce