I have a 20 amp breaker that is providing juice to the new outlets I installed in my basement. I would like to put a space heater in the three rooms. The space heater allows for (1000/1500/1600W, 120/208/240V) but I am concerned that 1000/120v will not be enough. Will I have to run separate wire for 240 or can I change out the circuit breaker with a 220? Will that impact what I can use with all of the outlets that are currently wired to the 20 amp breaker?
You would need to change out all the receptacles from NEMA 5-20 or 5-15 (125V) to NEMA 6-20 or 6-15 (250V) - it is a gigantic violation and hazard to supply 240V on a 120V outlet.
As such, running a new dedicated wire for the heater (if even needed) would appear to make the most sense, as you probably don't have many things with 240V plugs to plug in.
First, a habitable space requires a certain number of 120V receptacles that are mandatory. The rules vary by space, but typically you need a receptacle within 6 feet of any point along a wall (with the hypothetical 6' cord following the wall). That is because most appliances come with 6' cords. Whatever else you do, you cannot eliminate these mandatory receptacles and they must remain 120V.
If you change voltage, you must change ALL the receptacles
You are familiar with the standard socket that's everywhere. You must not use this socket for 240V. Not once, not ever. However there is another socket you use instead, with different keying so you can't plug the wrong thing into it. Other than the keying, it looks and installs like a normal socket.
Converting the existing circuit to 240V would leave you without any 120V receptacles at all. Therefore it is not a productive thing to do; you're better off adding a separate 240V circuit on a separate cable.
Cables contain multiple wires inside a single sheath.
Only 240V loads can be on 240V circuits
If you plug a 120V appliance into 240V, most likely you will get magic smoke. Possibly quite a lot of it. It's a thing you should not do. However if you install the correct receptacles on the correct voltages, that won't be anything you need to worry about.
There is no way for a 240V (only) circuit to support 120V loads.
There is a hybrid circuit called a multi-wire branch circuit, but it's a specialty thing that requires a lot of care to install correctly. It should not be attempted except by the most experienced DIYers, and recent rules about GFCI and AFCI make it problematic.
The simple answer is if your cable is 12/2 or 14/2 (black, white, and a ground) you can't power 120v and 240v from the same wires. If you have 12/3 or 14/3 (black, red, white, and a ground) you could make it work.
To be able to operate 120v and 240v on the same "multiwire branch circuit" you need 4 wires, which is two opposing 120v hot wires, a neutral, and a ground. The two opposing 120v wires would provide the 240v for the heater, and either of those hots (or each hot separately) could be used with the neutral to provide 120v.
A bathroom needs a dedicated circuit so you will need some new wiring.
I would suggest installing a sub panel (depending on the location and fill of your existing panel). I would want a 50 amp sub to provide power for the bathroom, receptacles / lights and heat. You could get by with less but when putting a sub As you have found not having enough to do what you want is a bummer. If your panel is full pull the 20 amp double pole and get a 50.
6-4 wire will be need and a sub panel.
You can get a panel of a larger size and it can be a main lug only but needs to be 50 amp or larger. The ground bus must be isolated from the neutral and most main lug panels come this way, if you get a panel with a breaker the bonding screw or jumper if installed will need to be removed. This is what I would suggest and that allows for a few additional circuits in the future. Edited I don’t know where I came up with bathroom, may have mixes 2 questions. If using electric heat I would suggest cadet built in electric heat. These are inexpensive and will last longer than most space heaters, I think Harper would agree with that.