It's really hard to answer that. You have to understand that the Lake District has a few tarmac roads and a mass of gravelly rocky byways, which evolved over centuries for horses. If you want to avoid the roads you will be on surfaces that can vary from easy smooth compacted ballast in the more frequented valley bottom areas to seriously rocky and steep bridleways over the mountain passes. The kind of bike you need depends largely on your ambitions. That said, before MTBs were invented people rode ordinary roadsters over Lakeland passes and I've seen blokes on road bikes up on the tops riding impossibly rocky trails.
One thing you won't encounter is lots of mud because the byways are generally firm and rocky unless you venture onto peaty areas.
Just a few miles before you enter Windermere from the south you pass a village called Staveley, which has a very nice cyclists' cafe called Wilf's:
http://www.wilfs-cafe.co.uk/ recommended. Also recommended in Windermere is Booths supermarket next to the station,which has a great cafe and a stone's throw away is the fascinating Lakeland store, which is full of solutions looking for problems and also has a nice cafe with good service. Try the Windermere boat trips, they're good fun; take binoculars for spying on the wealthy residents around the shore. If you tire of cycling, you can hire boats or canoes on Windermere or better, on Ullswater just over the pass.
A really great day out on a bike is to park at Hartsop then walk/ride/carry up to the top of High Street then ride north to Pooley Bridge, turning back south west before you reach the town and following the shore of Ullswater back to the start point. It's a tough day out, about 28 miles if I recall and takes you through some stunning scenery with a couple of cracking fast downhills thrown in. I can email a route map and some pics if you PM me.
