Do you have an external drive available?
If I was in your position, the VERY FIRST THING I would do is create a bootable cloned backup of my old Mac using either CarbonCopyCloner or SuperDuper.
You can then take this drive and connect it to the new Mac, and everything will be there in front of you in "plain ol' finder format".
You can copy one file or folder, or many.
You can run Migration Assistant using the cloned drive, if you wish.
Are ALL your photos within the iPhoto library on your old drive?
Is ALL your music within the iTunes music folder on your old drive?
If so, it's possible to just copy these libraries from their respective locations in your old home folder (on the old drive) to the new home folder on the new drive.
You will then have to hold down the option key when you launch iPhoto on the new drive, and iTunes as well.
IMPORTANT: will you be moving from iPhoto (older Mac OS) to the new Photos app in El Capitan?
Photos may need to create a new library for itself when you first launch it.
Mail can be more difficult.
It -can- be done, but involves moving specific folders and files within your home folder.
Or -- you can go to the old computer, open Mail, and choose to "export" individual mailboxes. If you have numerous mailboxes, may involve numerous export operations.
Then, import them into your new Mac Mail.app.