Where should you visit in Montenegro?
Firstly, Montenegro is the of the most beautiful country in Europe, another one is Bosnia and Herzegovina. As a person that have been to many countries, I was really surprised how Montenegrins are welcoming people. I can't stress enough!
Ok, coming to main question "Where should you visit in Montenegro", I guess you will fly to Montenegro - plane will land in Podgorica airport because it's only airport in Montenegro. There is only few places in Podgorica worth to visit - I wouldn't stay there more than a day. Maybe you can go shopping in downtown if it's not weekend. There is a park when on the left of road going from Old Town to Downtown called Kralyev Park, you might want to spend some time there, that's all.
In case you are not interested in shopping you can go to Bar city from Podgorica, best option is to go by train, it's in Podgorica very close to Downtown or you can take a taxi from Podgorica to Bar (even from airport). Taxi shouldn't take more than 30 euros, you should negotiate with driver. You should take a look at timetable in advance if you want to by train. Buses also go from Podgorica to Bar periodically.
Bar city is one of the best cities in Montenegro, There is a lot of to see there: plenty of nice and clean beaches along Bar Riviera. Also don't forget to visit Old Bar - very beautiful touristic place with a lot of historical buildings. If you are like swimming you can visit Ulcinj - only sand beach in Montegro close to Albanian border or Sutomore - beach and touristic market on the way to Petrovac.
I would advice to visit Petrovac, you can take a ferry tour from Bar visit Petrovac, as well as Canj and Queen's beach - they have tours each morning during touristic season from Bar port. Again a lot of history and beautiful beaches, as well as world class hotels in Petrovac.
Next on my list is Budva city, it's best developed touristic infrastructure in Montenegro, a lot of good restaurants (Particularly one with Biker Theme which I have forgot name). Also there is old town which worth to visit.
The best destination for me in Montenegro is Kotor city - situated close to Dubrovnik (Croatia), The water in Kotor bay is the cleanest what I have seen in my life. Also most of beaches in Adriatic sea have blue flag. Also Old Town of Kotor looks magnificent, a lot of shops, restaurants. Kotor having really high density of tourists because each day couple of big cruise liners arrive to city, especially there are a lot of American and Japanese tourists.
Also there is Durmitor National park, but I haven't been there. If you like more Nature than you can visit there as well.