Welcome to the forum iSteve
Generally, you will get the best quality advice if you include your current and former machines models and specifications. Good thread title though - wish every new member was as clear in their titles.
Moving data from one Mac to a new Mac.
The easiest way by far if your previous Mac was on OSX is to use the Setup Assistant that is built into the new Mac's OSX setup -- you connect the two Macs by a Firewire cable, and put the old one into Firewire Target Disk Mode by rebooting the Mac and holding down the T key until the firewire logo appears on the screen. This transfers your OS X identity, mail, documents, and everything in your User folder.
If your previous Mac is OS9 and has Firewire:
You can do the same Firewire Target Disk mode, after you have OSX set up on the new machine, and the old one will appear on the desktop just like a firewire hard disk. You then move your materials to the new machine by dragging them onto your new drive. I suggest setting up a "From Old Mac" folder to put them into first, so you don't accidentally overwrite anything on the new Mac that you should have kept. PS The Library, System, User and other folders that OSX has installed on your drive (and the respective folders within those) should NOT be renamed, overwritten or moved. OSX expects to see things in their correct places.
You can do file transfer over an Ethernet network if your old Mac is not within Firewire-cable range or doesn't have Firewire. Or you can string a Cat5 Ethernet cable between the two machines for a one-on-one network. You have to take the additional step of creating a shared folder on the new machine to dump the info into from the old. This is easiest if you turn on Appletalk first on the new machine under the Network Preference Pane: Built-In Ethernet settings.
The tricky bits are putting the things in the right places. This depends mostly on which programs you have - Mail->mail, web browser-> web browser etc.. That's too much to go into here, but there are lots of resources on the Web,
Mac OS X 10.3: Transferring data with Setup Assistant frequently asked questions (FAQ)
http://docs.info.apple.com/article.html?artnum=25773
Mac OS: Moving files from your older Macintosh to a new one
http://docs.info.apple.com/article.html?artnum=25575
In general, your documents should go in your Documents folder in your home folder, and other programs like Mail have some import routines if you have to bring in data. If you are going OSX to OSX you can often get away with putting some of the special data files (Calendar, mail) directly into the Library or User:Library folder places they came from the old machine.
You can also search within the MacRumors forum where this has been discussed before:
https://forums.macrumors.com/threads/123501/
https://forums.macrumors.com/threads/125292/
https://forums.macrumors.com/threads/122282/