Sounds like Blosson End Rot (BER), a common problem affecting Tomatoes. The link below discusses the causes and some corrective measures.
The first fruit set on Tomatoes is often small, haven't found any solution to that. Maybe someone knows the answer.

Here is a link that might be useful: Blossom End Rot (BER)

One thing I didn't see mentioned in the link that has always worked for me....

When you first plant the tomatoes, put about 1/2 cup of Lime (like you use on your lawn) in the hole and mix it in. It provides a substantial amount of calcium.

There is also a great product called Stop Rot. It is a liquid which is more easily taken up by the roots at this stage of the game than regular garden lime, which really needs to be added to the soil at planting time. You can find it at most "big box" garden centers.

If you have any scrap gypsum board (not painted) laying around, it is wonderful for crushing up and putting into the holes before planting as well.