There are a few known issues that you may encounter when using Time Machine. Below are some of the most common with any work-around or solutions if any exist.
If you use Microsoft Entourage as your email client, Time Machine cannot backup the Entourage Database if you have Entourage open. The easiest solution for this is to just exclude the Entourage database from the Time Machine backup. If your email is also stored on a server, for instance if you are using an Exchange or IMAP email server, then if you ever needed to restore your mail, you would just need to do it from the server.
If you do need your Entourage database backed up, the best alternative is to use third-party software that allows you to schedule the Time Machine backups. You can then make sure you don't leave Entourage running and then schedule a backup to run at night. The drawback of this solution is that your computer won't be backed up nearly as often as on the default one hour intervals.
Another thing to be aware of, is that the initial backup can take a very long time depending on how large your internal hard drive is and how much storage is in use. If you know you have a large amount of data on your drive, a good idea is to start the initial backup in the evening and then let it run overnight to complete. Otherwise, if Time Machine is trying to run the initial backup while you are trying to work on the Mac at the same time, you may notice a considerable slow down.
Also, the initial backup can be stopped and when Time Machine runs again it will pick up where it left off. It won't need to start over again. So you can also start the initial full backup, but if it seems to be impacting the Mac's performance while you are using it, you can simply stop the backup at that time and it will automatically start from where it left off at the next scheduled interval.
Time Machine works with many different file systems so most external hard drive will work right out of the box. However if you have any problems, the first thing to try is to reformat the drive. In the "Utilities" folder on your Mac is a application name "Disk Utility". Using Disk Utility you can reformat the external drive. If you are only going to be using the drive for Time Machine, I would recommend choosing "Mac OS Extended (Journaled)" as the format type since it's the native Mac file system.