It sounds like you do not have a new place yet. If so make sure you list your priorities when you go looking. And one of those being the ability to grow plants. Balconies are great if they face the right way and the angle of the roof is such that enough sunlight gets in. Same with the windows but as you look at the windows think about furniture arrangement. It won't do much good to have the most sun comming into a window that is also the only place the couch will fit. With that in mind what can you grow-----
I haven't found much that cannot be grown in a container with a little management. Tomatoes can be done in hanging baskets (buckets make good inexpensive pots) or plant smaller determinate varieties or there are some cherry types that only get a foot or so tall. Cucumbers and vining squash can make a lovely hanging basket or trailing vine around a staircase,window or balcony railing. Same with pole beans. Leaf lettuce can be grown in standard windowbox type pots (plant at least two a week apart for continuous supply) and if the window sill is not deep enough you can use shelf brackets and a board or wire shelving to extend the width of the window sill enough to support the pots. Carrots, beets, radishes,etc can't also be done in these boxes just pay attention to the depth needed. Peppers are wonderful in pots. Even with my large garden I usually keep some peppers in pots to overwinter out of the cold so I can get a head start the next year with them (peppers are at least biannuals this way). Most herbs do very well in pots.
Also if you are in a townhouse or ground floor place you may be able to set pots outside or have a spot that could be planted. Check with the landlord- they might be interested in turning a flower bed or two into a community garden if you manage it. Mixing and matching vegis and flowers can make a beautiful and useful space.
Hmmm just had a last thought if you are growing indoors be sure to check if the plant is self pollinating or needs insects. If self you'll be ok. If it needs insects you are either outta luck or you have to become a bee- it can be done - I have use a qTip many times to pollinate my lemon tree when it decided to bloom when I brought it in for the winter with good success I might add.
Dont think everything has to be right in front of a window either. Some of my best sun in my house is in the middle of the room. Might not be where you want to keep something but it could be moved there say while you are at work.
And don't forget combining things such as one upright growing plant along with something that trails or hangs down.
Good luck and don't hesitate to try something. Seeds are generally not way expensive so you only waste a little effort if it turns out a flop.
If I do not have a balcony, is there any system that someone could recommend that does not take up a lot of space? Something that is pretty cheap and cat friendly would be AWESOME!!! 
Thanks
