You can install the latest version of the Steam client on your new computer from here:
http://store.steampowered.com/about/?snr=1_4_4__11
... and then proceed to log-in to your existing account. You will need to verify your new computer if you have Steam Guard set up, which will include a code that is emailed to you that must be entered into the Steam client to access your account.
Once logged in, your library of games and software will be available to install.
Saved games that are not saved in the Steam cloud will not be on the new computer, and you may lose progress in games.
To save bandwidth and time, you may wish to backup the games you play most often and wish to install on your new computer. This way, instead of re-downloading them, you can simply copy the backup from the old computer, by way of network, external hard drive, thumb drive, etc., and just restore the backup to the new computer. This doesn't transfer save games, but it does save time in downloading.
To backup or restore your games, either individually or several at a time, in the Steam client, open the Steam menu and choose "Backup and Restore Games" and follow the prompts given.