Entering through the main entrance to Richmond Park, turn immediately on your left and after about a hundred meters, you will find a large parking lot. Here you'll find a bike rental company and small cafe, serving mainly burgers. In terms of the bike rental, you can get everything from kids' bikes to tandems. It's a great way too see the park!
Pier Head is the western starting point of the famous "Trans Pennine Trail", a hiking trail that can be done on foot, by bicycle or on horseback, to fully cross the island of Great Britain. It stretches from the mouth of the Mersey River in Liverpool to the North Sea, some 300 miles away. But you can just complete the local portion of the route if you want, although it's not easy with the strong winds!
The "Manchester Cycleway" is a network of bike paths that make a complete tour of the city of Manchester. The section that passes through Levenshulme (M19) is called "Fallowfield Loop" and is actually built along a stretch of disused railway line. It stretches from Salford in the south to Ashton in the north. Walking along this path will lead you to Highfield Country Park, and is quite enjoyable because of the works of art installed along the way.