The volcano, the waterfalls and the botanical gardens north of Hilo on the scenic route are the top attractions on the east side in my opinion.
The hike down to Pololu Valley north of Hawi is beautiful, won't take all day and is easy enough for most people. The upper highway between Kapaau, just north of Hawi, and Waimea is kind of neat as it offers great views and some different landscape.
Food... most for least award goes to Hawaiian Style Cafe in Waimea, it's basically plate lunch but it's very good too. To get a great meal that would run 40-80 bucks at the resorts, for about 15-28 bucks, head south to Kealakekua and try the Keei Cafe.... yummy. In Kailua try Splasher's across from the seawall for a good selection of pretty good 10 buck burgers, I've heard their breakfasts are very good also but haven't tried it yet. For a good hole in the wall spot I've heard good things about Rapa Nui, which is near Splasher's in the Banyon Court mall on Alii drive. For beer, if you are into it, go to Kona Brewing Company, good sandwiches and pizza too. The best overall meal I've had on the Big Island is the Surf, Sand and Stars buffet on the beach at the Four Seasons on Saturday nights, it runs in the 82 bucks per person range but it's good enough I try to make it once a year if I can.
For coffee, if you are in Kailua and want to sit across from the water and have a cup, try Lava Java. If you want to take some home with you, head on out to Greenwell Farms between Kealakekua and Captain Cook and sample what they've got, their top end Peaberry is over 30 bucks a pound but there are a bunch of good ones there.
There's so many things to see and do on the island you won't be lacking for things to do.
Have fun!