Could I ask what host you are using? Or is this a locally hosted site? Most big hosting companies (Bluehost, GoDaddy, HostGator, etc) have their own file browser that you can use within your web browser. Be forewarned though.. some of them quite frankly suck. I got the slowest transfer speeds in the world using GoDaddy's standard file transfer within my control panel there. Which is why...
I really like WinSCP for this. SCP is basically the same as FTP, but it is more secure, and allows you to do more. Here is the site for it: http://winscp.net/eng/docs/introduction.
And the differences (SCP vs FTP):
SCP: encrypted, including username/passwordd; part of ssh suite, uses port 22; can put a file anywhere you have privelege to.
FTP: plaintext, inc user/passwd, uses ports 20, 21; usually restricted to an ftp dir on server.
The caveat being that to use SCP, your servers both need to have SSH installed. Luckily, even if it does not have the SSH suite installed, you can usually ask your host to install it, or you can use FTP within WinSCP. To use this, you are going to need to know the IP addresses of your servers.
When you open the program up, you'll click on "new" in the top right corner. Here you put in the IP address of the server with your media, and decide whether you want to use SCP, FTP, or SFTP. You also need to know the user name and password for your sites SCP, SSH, or FTP access. This information will be given to your by the host in either an email, or it will be somewhere in your control panel. If it goes through, you will be presented with two file structures. On the left is your local computer, on the right is the remote. Simply navigate to your pictures (probably in /wp-content/uploads/), select the ones you want, and drag them over to your local computer, preferably in their own file. There is no picture preview here, so you may just want to copy everything, and then sort out which ones you want to upload later.
Once you are happy with your picture selection, log into your new server the same way - IP address, user / pass. You can now upload everything that you want to the new uploads folder.
This gets more complex with plugins like NexGen Gallery or similar media plugins, and for that you will just need to dig around to find your answers.