can I just plug in my old SSD in new mac?

Correct. macOS Extended Journaled is HFS+. You should not need to do anything, and it should just work. File system formats available in Disk Utility on Mac - Apple Support

Also, macOS can read and write to FAT and exFAT just fine. It's only NTFS that poses the "writing" issue. It can read from it, but cannot write to it.

Assuming its an external SSD, it should work fine just plugging it in if its already formatted to something the Mac understands natively like HFS+, FAT, exFAT or APFS. If it is formatted as NTFS and you were using its own software to access it, then you may need to install that, or some other NTFS driver for the new Mac to be able to write to it. It should be able to read from it however without additional software.

Make sure, especially if you're using a pro app like Logic, to have the drive formatted to HFS+ or (I highly recommend) APFS. NTFS , FAT, and exFAT your Mac can read just fine, but writing is inhibited and they won't be fast enough for a pro app.

Although FAT and exFAT is read- and writeable on mac, it's not recommended to use with Logic except for file transfer. Working with Logic directly on Fat-family formatted drives can lead to file corruption.

Like Phil0124 said, you're good to go, that's it.