If you have a hill to climb, a lighter bike would be an improvement in your effort level needed to get to where you want.
If you are comfortable seated upright - a hybrid or mountain bike may be better for you, if 'maximum speed' is not your end goal.
A lot of people say good things about the Carrera Subway from
Halfords - however, almost 14kg on the bike alone mean climbing is not as easy as on a 12kg bicycle.
If the road is often muddy - road tyres (or thin tyres) may not be the best solution unless you are comfortable on a bike and confident of not falling off.
Generally, it depends what you are most comfortable with and would be happy to ride. Something with a bit more pace but capable of on and off road would be a hybrid bike - they vary from £150 to over £500; a mountain bike is a good alternative but often is harder to go as fast as a hybrid.
Many people favour hybrids as they offer the flexibility of a mountain bike (i.e off road) while 'go faster' than a traditional mountain bike.
A road bike is a good idea if you want a 'sporty' bicycle - but bear in mind that offers little comfort in place of emphasis on speed. (But can be customised to your liking - as can all. But, generally, a light frame is a good place to start as it makes any cycling easier).