This may seem like a crazy question, but I am currently trying to design a kinetic machine where pilots are facing away from the direction of motion, adding whimsy and curiosity. As someone who doesn’t have a lot of experience with redesigning bicycles, I am struggling with coming up with a functional design where the gears of the bicycle could still be changed. The first idea I had was to arrange the chain in a figure eight shape, making the back wheel of the bike turn backwards while still pedalling forward. Although I haven’t prototyped this design, something tells me that having the chain cross like that would create too much resistance to reliably change gears. Another idea is to add an extra gear between the pedals and the chain, changing the direction the chain is going entirely. However, I’m not sure if the size of said gear would interfere with gear ratios on the bike? If anyone is interested in helping me out, I would greatly appreciate it!
I refer to a bike throughout, though I suspect you're actually going for something with more than two wheels, and likely more than one rider.
Multi speed bike gear systems generally can't be driven backwards. This means that if you really want gears, the chain that drives the rear cassette or hub gears must move forwards. Fixed gears are designed to be driven backwards for braking.
You could easily choose to go for fixed gears on a machine like this, in which case your figure of eight chain would work, but you'd want a chain tensioner with guides in the slack side, so the chain doesn't touch itself at the cross.
If you really want a range of gears the best thing would probably be to install a geared hub the wrong way round. This would need a left side chain, but you should be able to achieve that with tandem or BMX components (tandem parts are more likely to be compatible with other types of bike).
This approach would leave the bike as effectively a front wheel drive, rear steer machine, which would be very unstable. You might be better off driving the bike as normal (rear wheel drive, front steering) and turning the rider round. This would be (conceptually) as simple as clamping a saddle onto the top tube and putting your figure of eight chain in place as above, but would work with normal derailleur gears as the chain would be pulling correctly. A single chainring and a couple of idler sprockets with guides would be a good idea to maintain reasonable angles in the vertical plane.
For a little inspiration you might want to read up on the retro direct drive. This had one gear ratio pedalling normally and another driven (still forwards) by back pedalling, and the bike could still freewheel fowards.
Your point about intermediate gears is easily dealt with by making both gears on any intermediate axle the same size.
Going with some recumbent designs, have your drive train as usual with a bike but instead of driving a wheel you drive a cogwheel, from which you can drive your wheel or whatever you used to drive your 'bike'.
Do you know about the Human Powered Vehicle Association?
They have many more vehicles than I did see in the site. Some of them are back to back tandem recumbent cycles, where one of the riders sits back to the drive direction. For those they have to worked out how to switch the direction of leg movements compared to direction of travel.
(Link to a drawing of one of those tandems.)