Having spent a couple weeks in the UK going from place to place, I found that the National Rail Website was extraordinarily valuable to plan rail trips.

Also, while Gatwick Express is useful, I was also able to find fairly frequent service from Gatwick to London Bridge. Thameslink is worthwhile to use if you're travelling off peak and light.

The only caution on the National Rail website is to pay attention to the connection modes and timing. Some connections are "Tube" or "Foot" and I noted that it will sometimes offer a connection that departs at the same time as your arrival. Since many stations have multiple platforms, it's unlikely that you'll make a tight connection. Overall, I found that I could plan journeys around London that were comfortable and convenient.