Alicante is a definate must if you're going to be on the east coast. I spent a month there last summer. In addition, there are many small cities around Alicante that are great for sightseeing.
If you're going to spend some time in ALicante, try taking the ferry out to the Island of Tabarca. It's a small island that used to be used by pirates many years ago. Plus it's in a marine reserve, so snorkeling around the island is great.