If they are high definition (Hi8 and MiniDV),
you would want them transferred to BD.
Standard definition (for example, VHS that isn't "Super")
will come out OK on DVDs.
Or, you could have all of them transferred to BDs
just so everything would be on the same format.
Another option is a hard drive.
You can store copies of the material in the "cloud", but don't make that your only backup.
It depends on servers that can be damaged or destroyed just as easily as anything else can be.
Always keep a physical medium in a safe place, and Keep The Original Tapes, Too.
They will deteriorate over time, but if you need another transfer and they are still playable,
you'll be glad to have them.
For audio, CDs are the best, if only because there are so many players for them.
Remember that DVD and BD players are required to also play audio CDs,
so there are a lot more than most people realize.
With the right equipment (I have a Philips CDR820 CD recorder
and a few working tape decks), copying cassettes is easy enough.
Professional transfer services Will Not copy anything that is copyrighted,
such as prerecorded video or audio tapes.
Since most of the music most people like is available either on CDs
or as downloads, it shouldn't be a problem to reacquire as needed.
The weird stuff that isn't and won't be available is tricky.
Most of the oddball material I have is on records, and I have copied some of it to CDs.