Ok, so grab a tea and a biscuit... This is long, but this is how I would do it :)
Firstly look at how your furniture sits together, just because its all mid century doesn't mean it all works together. Has a chair got delicate legs whilst a sofa is a huge, monolithic block that sits on the floor? Some people can make contrasty things work together beautifully, but if you want to keep things simple try to make things complement each other, and have an edit if necessary. Does your furniture have similar 'visual bulk' in the room, do the shapes and colours and colours each other?
I'd then spend time finding a lovely neutral to paint the walls in, something that really complements the direction the room faces and underlying tones in your midcentury furniture. This can be hard to gauge if the furniture is predominately wood, but hold a sheet of white paper against it- do the undertones looks slightly orangey, or red or yellow?
That would give me a starting point. Lets say for example your furniture has a yellowy undertone and is south facing. In my imagination I've just painted it something like Farrow and Ball 'Wimborne White'. Its a neutral with a tiny hint of yellow and it enhances the light bright feel of south facing rooms. Next, if the room doesn't feel tied together and you don't feel very confident putting together a 'scheme' I would instead pick an accent colour or two that complement the yellow toned neutral that I'd established for the room and start threading touches of it through the room. A dark teal blue and mustard yellow can look good with mid century, but you can pick anything really! Personally I think its best to weave it through quite subtly, maybe find some artwork with the colours you like, then a cushion in that colour, then a neutral coloured curtain with your second accent colour as a pattern- you get the idea:).
So at this point the room is predominately neutral with your one or two colour accents. My next step would be to add lots of texture in the room- for some reason this just really seems to help. So say, a knitted throw, a slubby linen blind at the window, a wooden bowl, a seagrass basket, things like that. You don't have to go overboard, a couple should do the trick. Then if I haven't done so already I'd add houseplants and artwork, it seems incidental but it really does make a huge difference. It doesn't have to be fancy- a plant from the supermarket in a nice pot, a picture you've bought in Ikea or from Etsy (remembering your accent colours of course!). If the room is still not working, the room probably needs a touch of black. It can be as simple as a black picture frame, or a black lampshade. I've no idea why this works, but it just does! It makes the room look more 'finished'.
Good luck! And take your time and enjoy it! Think of building it up in stages or layers, rather than trying to do everything and find everything at once.