I think you need to refine your requirements a bit. Transmitting Morse code isn't that hard. This is usually done by keying the transmitter on and off. That's what some people erroneously refer to as "CW".
The reason you hear beeps at the receiver is due to how the receiver works. The transmitter is either sending carrier or not sending carrier. Making beeps was usually done by mixing the received carrier with a locally generated signal of just a little different frequencies. The two would mix, or "beat" against each other to produce the audible tone. When the carrier went away, so would the tone.
For example, let's say your transmitter is sending bursts of 27.0000 MHz for the dits and dahs of the Morse code. If the local oscillator in my receiver is set to 27.0004 MHz and mixed with the received signal, a beat frequency of 400 Hz will be generated. I will hear this as a beep every time you send a dit or dah. If I wanted to hear your dits and dahs at a different frequency, I'd adjust my local oscillator accordingly. There is nothing you can do to control the frequency of the beeps I eventually hear.
For a simple homebrew project just to learn and have some fun, I'd do the transmitter as I described. Find some simple RF oscillator circuit and have it enabled when the key is pressed. The whole setup will be hard to debug without a lot already working, so it would be useful if you can get hold of a shortwave radio. Some of those will have a keyed transmission mode built in with the local oscillator creating the beat frequency as I described. Then you can listen to your Morse code transmissions and verify that the transmitter is working. If you can find a old "regenerative" receiver, it will have the ability to make a beep when it receives a certain carrier frequency built in as a inherent part of its operation.
Don't try to go for long distance or high power, especially at first. A transmitter you cook up yourself will likely cause interference somewhere, even if you intend it to be on a band where something is allowed. If you do too much of this, you'll piss off someone, and if they complain you can get into trouble. It helps if you live someplace where the neighbors aren't too close. If you keep the power low and the signal from spreading to where anyone else will care or notice, then you can use whatever frequency you can manage to create a transmitter for.
Eventually you can try around 27 MHz. It's been a while since I looked at the regulations. That is where the old CB bands were. You could buy "walkie talkies" for those bands that you could use without a license at all. The range on those was usually under a mile, but most didn't put out as much power as they were legally allowed. Lots of rules may have changed since the last time I looked. I have no idea whether the CB bands are even still there. If I remember right, Morse was actually forbidden on those bands, as they had to be normal AM voice. But again, keep your power low, antennas small, away from where people might care, and it's very unlikely anyone will come after you.
