Do I need both a UK visa and a Schengen visa?

Yes, you do need two visas, since the UK and Schengen have separate immigration arrangements.

While the United Kingdom is a member of the European Union, it is not a member of the Schengen area, nor is it subject to the Schengen Visa policy. The UK is part of the Common Travel Area with the Republic of Ireland, however travellers should be aware that while immigration controls are not ususally in place between RoI and the UK, holding a visa for one of these states does not necessarily permite a visitor to enter the other. The below (slightly complicated) image indicates various overlapping European insititutions, some of which (Schengen, Eurozone, CTA, Customs Area) are relevant to travellers:

(Image CC-BY by Wikipedia user Aris)

Given that the UK is not a part of Schengen or bound by the visa policy, Schengen Visas do not count as leave to enter the UK. The UK's government website does provide a useful tool for checking if you need a visa (and if so, what sort). It does not cover all possible cases, but covers most common situations and exemptions that a traveller may have.

A Schengen visa does however offer the possibility of entry to some non-Schengen states. Bulgaria, Croatia, Romania and Cyprus are all bound by the previously mentioned Schengen visa policy, and should give access to those holding a valid Schengen visa. Although as this involves leaving the Schengen area to do so, the Schengen visa must still allow reentering the Schengen area (otherwise the validity has now expired). Wikipedia also has a (mostly unreferenced) list of other countries for which Schengen visa will allow access (often with additional requirements)