Both types of mixers have their place in the kitchen. A hand mixer is great if you are making batters. Some hand mixers are strong enough to tackle cookie dough, but be prepared to spend at least $50 on up for such a mixer.
A stand mixer is far more versatile, does more than a hand mixer, and cleans up just as easy, if not easier, than a hand mixer. I own both, but if I had to reccomend just one:
Hands down, a Kitchenaid stand mixer. Why kitchenaid? It is absolutely the best for several reasons. Kitchenaid stand mixers are durable, and last for years. I've heard of people who have had their kitchenaid mixers well over a decade, and used them regularly. You get a wisk attachment, dough attachment, and batter/cookie dough attachment. This stand mixer also has optional attachments to make pasta, grind meat, make icecream, and do much much more. The driveshaft is accessible via a screw on the top front of the mixer, so you can add a myriad of attachments.
I am biased with kitchenaid, as I had amazing results with mine. My experience was so good that I now swear by kitchenaid for appliances. I have my suite of empire red kitchenaid gadgets, including blender, 12cup food processor, immersion blender, in addition to my stand mixer. I held off getting the dual beater hand mixer due to my old one working perfectly, and couldn't yet justify the empire red toaster or toaster oven. :P my toaster and toaster ovens work fine, thank you.
stand mixer all the way! It is pricy, but well worth the investment.