Access files stored on Time Machine

Time Machine is pretty much all or nothing when moving from an old to new Mac.

If you only want some files from the old Mac, then your best bet is to just copy those files to an external drive or make a clone of the old Mac's that you can use in Finder.

At this point some options include:

1. Migrating everything from the old Mac's Time Machine drive and then delete the unwanted files from the new Mac.
2. Manually copy what you want from the old Mac to an external hard drive or flashdrive and move them to the new Mac.
3. Connect the Mac's together in Target Disk Mode or by using Network Sharing and move over the files that you want.

Obviously 2 and 3 would require that the older Mac is still working.

For more on moving to a new Mac, see > How to move your content to a new Mac - Apple Support

I want to be able to access a select number of folders on Time Machine but I get the message along the lines of 'you don't have access to the backed up files'.

Are you doing the above via the Finder or while in Time Machine? You should be able to enter Time Machine, find the latest backup of your desired folders and restore them. If your TM app can't access your TM drive try holding down the Option key and then access the TM menu:

If you have trouble accessing that drive in Time Machine look at Section E of Time Machine - Troubleshooting