I need to install a bike chain for my 10 speed

It is best to use a chain tool to press the pins in and out. They will prevent damaging the chain when pushing the pin. Chain tools are cheap and totally worth the money because you won't have to replace the chain again due to damage.

Cyclo Rivoli is the cheapest chain tool and works quite well, sold all over the place. Here is an example: http://www.amazon.com/Cyclo-Rivoli-CY-158R-Chain-tool/dp/B0009W81NM

Hi
very often you need to replace the rear cluster at the same time as the chain, as it wears to the shape of the old chain and when a new chain is fitted to an old cluster it can sometimes jump, which is not fun if you are standing and pedalling up a hill ( not quite as bad for the ladies)
anyhow remove the old chain by punching a pin out of a link, and lay it on the ground next to the new chain, punch a pin out of the new chain at the same lenght.
Thread the new chain around the cogs and derailleur and fit the joiner, there are some really neat joiners that will just slot together. or you can use the link from when you sized the new chain, just press the pin back through with multy grips, make sure the joiner link is not sticky at all, or it will jump when it goes around the cluster