I'd stay in Seminyak/ Jiimbaran/ Canggu area whenever i'm there. It's close and convenient to places to go and yet not too crowded. Ubud is also nice if you want to spend more than one day to explore that area (paddy fields and all - we once rented a car and just drive around the area and found some magnificent view of paddy fields. But you can also rent bycicle or motorbike to do it).
Temples to visit, "Pura Besakih" being the 'mother temple' - biggest and holiest. Others that i like: Tanah Lot, Uluwatu.
Batu Bulan, Celuk, Mas, Ubud for the arts/ cultural thing.
I'm not into hiking, but heard that you can do it at Munduk(?). Maybe someone else can help this one.
Window shopping and furniture shops - you'll find it along the kuta-legian-seminyak...
Places to eat (i'm adding this because that's what i like to try

) - for breakfast "Cafe Moka" in Seminyak is a must for me (small place). Price is incredibly cheap for what you can get.
"Murni" at Ubud - their smoothies is delicious. "Warung Made" at Kuta - the old one is more original.
By the beach dining - La Luciola (italian), Jimbaran bay (for seafood).
Others worth mentioning: Trattoria, Kafe Warisan, and lots of other eating places... (can't help it - i love food).
There's still more to do... Enjoy!
PS. when you're walking along the beach or shops, just ignore the peddlers, otherwise you'll be stuck with them. walk straight, not encouraging them. With Bali being pretty quiet these days, i'm sure the peddlers will be 'hard at work'....