You can travel around the EU from the UK as you would coming from the US; there isn’t any difference. Just remember you are travelling on your native country’s passport and your UK visa won’t mean anything in another country.

It is advisable to check if any country you plan to visit requires a visa for entry. Egypt, for example, requires a visa of all visitors. Check this before travelling, as well as being sure to carry all of your documents with you, not just your passport.