Okay, so let's be upfront about this: Negara isn't geared up to be a holiday destination, nor does it pretend to be – but there are a few places around the (very small) city that make for an interesting day trip.
Perancek is a fishing village about half an hour's drive south-east of Negara, or an hour's drive west from Medewi – and so worth a visit if you're in the area. The boats here are unlike any others on the island: they're big, bright and always work in pairs, so you'll always see two decorated the same and moored together.
On Sunday mornings between July and October, the villages and paddies between Negara and Perancek become covered with dust and excitement: it's at this time of year that the traditional mekepung – buffalo chariot racing – takes place. Two ornately decorated buffaloes pull a small cart driven by one man, and races happen two-by-two throughout the morning. The later in the day the more bloodier the races (because the mekepung drivers wield nail-tipped koshes), so if you're sensitive to animal rights but are interested in watching the races, get there early. Read more about the races and traditions here.
About an hour north-east of Bali you'll find Palasari, the island's only Catholic village. The cathedral here is beautiful, and the drive up to the village is really scenic.
As the biggest city in this part of the island, Negara is a good place to go if you need to stock up on supplies from a supermarket (Hardy's has everything from clothes to cosmetics to beer to appliances). All of the major banks are in the city, and there is a good hospital here, too.
Negara: local advice
...if you drive
The road to Negara is the same one that connects Gilimanuk, the port town close to Java, with Denpasar – and so it's usually packed with (often old) slow, overloaded lorries. Don't drive this road in a hurry, and avoid it after dark, when you'll be less likely to see the broken branches that are used instead of warning triangles when a truck breaks down.
...if you need cash
There are a few ATMs next to Hardy's supermarket where you can withdraw up to Rp2,500,000 (many other ATMs in this part of Bali allow a maximum of Rp1,250,000).
...if you need groceries
There are two supermarkets that we know and use in Negara: Clandy's (excellent for supplies for babies), which you can't miss if you look for a large building that's painted pink, and Hardy's. Both are easily found if you look on Google Maps. Know that dairy products (yoghurt, pasteurised milk and butter) are often not available. Hardy's has a small booze store where you can get wine and spirits. If you're staying a while in West Bali you might appreciate a delivery of groceries that are mostly organic and produced in Bali. Bali Direct is based over east but will deliver to Jembrana for Rp200,000.