Cerro Monserrat is a great place to start seeing Bogota- it really shows the size of this Andean city, although it can be pretty chilly up top! Bogota has an interesting touristy centre (north of the main plaza) where there are many restaurants and cafes. La Rosa area is good at night (safe) and has many bars/ nightlife.
Insider Tip:The Museum of Gold is worth a visit. The Emerald district is great- but only if you know your emeralds!! Read up before you go if you don't already know!)