Firstly, everything JB has said on french press is spot on, good advice to be had there.
For those looking into grinders: Krups make one that can be bought relatively cheaply (about ?30 here in the UK) that allows a good amount of adjustment of coarseness/fineness of the grind and has survived my girl and I's daily usage for about 2 and a half years now.
As for aeropress vs french press: I own both and hardly ever use the french press now. French press coffee seems ever so slightly more viscous, all delicious and velvety, but I often feel all jittery and overcaffeinated after 1-2 mugs which is a problem for me as I like to drink a few coffees throughout the day. Aeropress coffee is slightly thinner and it avoids the issues with mouthfuls of grounds at the bottom. It feels like an altogether 'cleaner' cup but it doesn't seem to lose flavour, just a different texture, and it is quick to make a cup & really easy to clean up after the fact. The other thing with the aeropress that I really like is how good it is to travel with; it doesn't take up much space in a bag for everything you need bar hot water and it's mostly indestructible, especially in comparison to cafetieres.