I agree with all of the above recommendations, especially the Fine Diner and Glen Gardner Inn. I'm not a big fan of Pru Thai, preferring the Thai restaurants in Somerville/Bridgewater area, but there are not many Thai options in the immediate area around Clinton. (There is also Savory, a Thai restaurant as you head south on Route 31 towards Flemington, but I've not gotten there yet so cannot compare it to Pru Thai.)
Sticking to the vicinity immediately around Clinton, I'd add the Grille at the Pittstown Inn to the list of places to check out. Its about 4 miles west of town on Route 513, a short nice drive into the surrounding countryside.
There are many good choices in Hunterdon County but for the moment I'll confine my comments to places very near Clinton or just to the north.
I do agree that the Fine Diner is a terrific choice for breakfast or lunch. It's nearly always busy but you rarely have to wait long for a table. Imho, the quality of food is very good and I especially like the place because service is quick and very friendly.
For dinner, someone has already mentioned Pru Thai, a somewhat above average Thai restaurant on Main Street. There is also a sushi restaurant on Main Street but its name escapes me. Dora, an Italian restaurant is nearby and I have always had good meals there though I must admit I usually stick to pasta dishes (I like pasta!). Dora also has a nice outdoor area, set back from the street, where you should sit when the weather is good. The Clinton House is just across the bridge from the downtown area and is a great place to go on a cold winter night. It is old school and can be pricey if you're not careful. However, the specials are usually less expensive and if they are offering sauerbraten on the night you're there be sure to order it. It's very good. You should also explore the Clinton House bakery, in the same building, that is open until 6 PM every night. It is take out only but they have some of the best pastries I've had anywhere.
North of Clinton, in Glen Gardner, is the Glen Gardner Inn, which I like because it offers small plates of nearly every entree they serve. I've never been disappointed there.
Gronsky's in High Bridge has already been mentioned but it is worth a visit, if only to have the pancakes! When the weather is good, just get an ice cream and sit at one of the outdoor picnic tables next to the river. Two other places in High Bridge (actually three) are also worth a visit. Circa has changed a lot since it first opened but it has a devoted following and offers many special events. I believe it's operated by ex-Ryland Inn folks. Casa Maya, just down the street from Circa, is a pretty good Mexican place that has a sister restaurant in Stirling. In the same building is Café Hola, a small restaurant that offers breakfast and lunch. Casa Maya and Café Hola are owned by the same people but the food at the cafe is more Cuban influenced. Try the Cuban french toast!
I highly recommend the Fine Diner, http://www.thefinedinernj.com/ on Main Street in Clinton, just a couple of blocks from the rte 78 exit/entrance (not to be confused with The Clinton Station Diner which was just meh the last time I was there). Great upscale food, not at all your typical greasy diner. Definitely a chowhound worthy place.
Heading north, there is a great place for cheese in Long Valley, http://valleyshepherd.com/ great local artisan products.
Another great resource is the PA Dutch Farmers Market in Flemington. http://www.dutchfarmersmarket.com/
You are in luck being that you are close enough to do your grocery shopping at the Wegmans store in Bridgewater. (though I recommend getting your meats at the PA Dutch farmers market...better quality IMHO)
If you are north of Clinton, forget it. Cook your dinner. In Clinton town you can get decent Thai food at Pru Thai or good subs and soup at Ye Olde Sub base. You can go down to Flemington for a better selection. 55 Main has excellent food, but it is a little too small and crowded for my taste. Matt's is nicer inside and the food is pretty good. Both standard american fare. For a big tasty breakfast , try Gronskys in High Bridge. Service is slow and the place is tiny, but it is good
It's about 20 minutes, but Frenchtown is a bucolic little village with some restaurant gems.
Frenchtown Inn is a glorious place for a serious dinner, or the bar area has a separate menu with a more casual environment.
Lovin'Oven is a great casual for breakfast or lunch.
I also hear good things about Bridge Cafe.