Does anyone know if there are direct buses from Munich to Bayreuth, or trains. Travelling with 2 small children and do not want to rent a car. Any advice?

There are frequent trains, but you have to change in Nurnberg.

You can save money with the Bayern-Ticket, but you are limited to the slower trains.

The slower trains to Nurnberg are not all that much slower. You are limited to regional trains after 09:00. The train from Nurnberg to Bayreuth leaves hourly and is scenic. Bayreuth is a pretty town, although heavily bombed during WWII.

I would not recommend to opt for the Bayernticket having to go all the way on regional trains. Bayreuth is some 220 km (135 miles) from Munich and travelling there exclusively on regional trains will take close to 4 hours - unless you are keen to experience how travelling in the olden days of horse drawn post carriages felt it's quite a waste of time in my opinion.
Taking an express train from Nuremberg gets the time down to a more acceptable 2.5 hours.

At second thought: If money is of relevance, you admittedly do save quite a bit of it (esp. since you are travelling with family) if you take the Bayernticket and its associated sloooooooowww trains.
If you are on a vacation with limited time and want to maximise your time exploring places, I'd tend to take the quicker option, otherwise it may be worth considering the daaaaawwwddling regional trains. If you do, make sure you take got lots of entertainment options for you and your kids, though.

There are no bus connections between Munich and Bayreuth.

For train connections go to www.bahn.de
All connections (slow or fast) require one change in Nürnberg.

If you book in advance, even the fast trains can cost only a little bit more than the slow regional trains.
The cheapest Saver Tickets for fast trains start at €29 (incl. your kids), the cheapest for slow/regional trains only start at €20 (if your kinds are both under 6), or €28 (if kids are 6-14).

There's one daily bus service to Bayreuth from Munich, operated by Berlin-Linien-Bus (www.berlinlinienbus.de).
The bus leaves from outside Munich station at 9.45am arriving in Bayreuth at 1.30pm. You cannot book this online, but have to call 09281-2252.

I would advice against it, but prices are rather reasonable if booked far enough in advance and you get first hand autobahn experience .

And I thought there had to be that bus, but thought they discontinued the service when it did not appear online.