Lebanon is a tiny country and Beirut is super central so it's easiest to stay there for the whole time and just do day trips. The other option is to stay in Tripoli one night and do Qadisha Valley from there there the next day. I stayed in Hamra because I prefer a more chilled scene, and I liked running by the Corniche in the morning, but you can also stay in Mar Mikael. I highly recommend my Airbnb host, Bassel, who is super hospitable, will pick you up from the airport, and share his country in such an earnest way with you.
The easiest thing is to go with with Nakhal Tours, their tour guides are expert and you get a feast for lunch, but it can be on the tad expensive side. Even if you don't go with them, their tour packages are really good itinerary for the things to do. To get a car for the day is a better deal if you are more than one. A day trip to Byblos, Jeita Grotto, and Jounieh/Harissa is about $80-90, while Tripoli is $120-130. I would really recommend Salim Taxi (whatsapp +961 3 290 493).
Data is super expensive, and a sim card with 1.5 GB data for 7 days will set you back $20.
Stay Up All Night
Ballroom Blitz is a great for clubbing (check their facebook page for tickets to their various special parties). In Mar Mikael, Bohemian is a bit pretentious, but fun to meet people. I prefer the drinks at Amelia's across the street.
In Hamra, Dar Cafe is cozy with a cute book store, and they also make a mean Gnocchi. Tales & Ales always have good vibes, Propaganda Gin Room is pretty new but has an extravagant Gin collection. If any of the places are full, the best bet is the next bar next to it is just as good. The special Lebanese Shot is called a "Do Do Shot" (vodka, lemon, tabasco, etc). Do it!