I think the small closet will be fine for vegging. I love T5s for vegging and would recommend something like this:
http://www.amazon.com/dp/B002TJQ61W/?tag=skimlinks_replacement-20 or this:
http://www.amazon.com/dp/B00HE04WTY/?tag=skimlinks_replacement-20,
You will need an exhaust fan--a 4" should do, but a 6" isn't much more. The one DGF linked would be great. You will need a speed controller for the fan. You need an oscillating fan to move the air around within the space. A carbon filter is not needed in the veg space.
It will take the plants about 6-8 weeks after planting to be ready to flower. During that time, you will need to get a flowering space set up--the other closet will probably do just fine. I really like using closets, but you do need to keep in mind that you need electricity (and adequate amperage) in there and you have to set up ventilation, which entails drilling holes in things. Can you do that? Are you a handy type person who can use a drill and a big drill bit? If not, tents may be a better option.
Have you decided on seeds yet? You should think about what kind of high you want and then you can find seeds that meet that criteria. There are so many hybrids and crosses that you can probably find something that is exactly like you want as far as taste, type of high, and growing characteristics.