Herceg Novi, although not the most spectacular city in Montenegro (Kotor takes that prize), Herceg Novi is probably the most pleasant and warrants a several day visit. The city is particularly a good alternative to the very touristy Dubrovnik in Croatia. The cities have similar architecture but Herceg Novi not being as grand (nor as touristy and expensive). Herceg Novi translates to English as "New Castle".
A sizeable Bosnian (Serb) refugee population flooded into Herceg Novi during the war years. Many of the camps are still around although they have been turned into more permanent (and quite nice) settlements. You can still see them on the bus as you leave the city. Also if you go for a wander up into the hills you will come into whole new areas of construction resulting from the recent property boom. The Financial Times in 2007 listed Montenegro as one of world's 10 top property hotspots and foreign buyers have been snapping up properties on the coast.
Bus is the only form of public transport in/out of Herceg Novi, and the bus station (located in the centre of town) is busy the day long with buses heading (mostly) down the Adriatic coast. There is a regular bus service to Herceg Novi (and further to other Montenegrin cities) from main bus station in Dubrovnik, which runs several times a day.
If Herceg Novi is your main destination, a pleasant alternative to bus travel is hiring a Croat cab from Dubrovnik airport (Cilipi) for about €30 .; this trip takes about 40 minutes, including border crossings. There is NO rail or ferry access into the town.
The Herceg Novi old town is amazing. It is on a fairly steep hill that leads all the way down to the sea. Wandering through the small stairways to the various plazas and fortresses is a many hour adventure. There are both Orthodox and Catholic churches that are well worth visiting.
Italian-made clothes in Old Town and Igalo boutiques are reasonably priced up-to-minute fashions. Not great for local arts and crafts, which are available in Kotor and Budva. Go to local market just off the main square in the Old Town on Saturday morning to buy fresh fruit (sweet and cheap), olive oil, sheep and goats cheese, dried figs, locally made wine; note that this market sells seasonal and locally produced goods, so what you find depends on the time of your visit.
Go to small eateries around the promenade for fresh local food, grilled seafood and meat dishes, and international cuisine. If you order pizza, which is usually well-made. Follow the locals; Montenegrins are usually more picky than foreign tourists when it comes to eating out.
Herceg Novi is hot in summer, and it usually stays warm late into the evenings, so cold drinks are best sellers. Iced coffee is served in tall glasses with dolops of ice cream and "slag" (low-fat whipped cream) and qualifies as a full meal. Local beer made in Niksic is good; also try "spritzer" which is a refreshing mix of chilled white wine and carbonated mineral water.
Accommodation is plentiful, look for signs that say 'soba.' You should be able to get a room for 10 EUR per person during the summer months.