I posted a couple of thoughts in a PM, but in the basis others may find it helpful, here's a slightly abridged version
(also so it doesn't look like I'm ignoring
@Jackslad)
As
@chriswoody says, carrying bikes on trains here is relatively uncomplicated. You can carry normal bikes on most Intercity (IC) trains, but you'll need to reserve a bike place.
Here is a shameless plug for a recent blog entry about travelling by train with my bike. You can see the bike rack is a bit tight. There are carriages with more spaces though. There are also a lot more low floor IC trains.
Unfortunately I've noticed newer trains are using bike racks that don't allow space for longer bikes such as my Xtracycle, so I have to use local trains.
Regional trains; Regionalbahn (RB), RegioExpress (RE) and InterRegio (IR) generally have a space for bikes, and are generally a lot cheaper than Intercity and ICE, but you can't reserve a place so you have to just get on board any old how.
Our state is investing a lot in rail transport and trying to encourage bike use, they're getting new operators and insisting they allow bikes on board. Some trains from the last week:
As you can see they are well used. Cyclists are generally pretty good at helping each other. Pushchair users on the other hand...
Platforms in rural stations are very low which means older trains without low floors can be hard to get your bike on because there are three or four steep steps to get up. In our state this is thankfully mostly just IR trains, but I don't know what they are using further north.
There are variations depending on the state you are travelling in as to whether you need a ticket, and to make things a bit more complicated occasionally there are variations within states.
As others have said the DB website is very good. (Works better than their trains) If you want to check I'd suggest the Deutsche Bahn website which works very well and has an English option. If you click on "further options" it gives a tick box for "Only show trains with bicycle carriage (Buy ticket if needed)" at the bottom of the page:
https://www.bahn.com/en/view/index.shtml
One thing I found last week is that when I included the metro (S-bahn, Ü-Bahn or Stadtbahn) in my journey and ticked that box it freaked out. If I just typed "Stuttgart-Aalen" it gave me a connection.