You can't go by color alone, you have to understand what the wires do. Here's what happened when you hooked all the like-colors together. BOOM.
This is probably how it was connected originally. However, this is wrong and illegal, for several reasons: white wires must be neutral unless marked otherwise, and here it's a hot mess. Also, "hot" is always connected to the light fixtures even with the switch off, which can surprise the guy working on the fixture. It's also likely that the white wires on the fixtures are actually "hot" - that's especially dangerous with Edison style sockets.
Here are the exact same wires, rearranged in a way which is legal and solves all of the above problems. Now the "hot" wire to the fixture (red) is switched, and neutral is always connected. With the switch off, the fixtures see no power. Note it now matters which end the switch is on, because on one end, white is actual-neutral, and on the other, it is switched-hot.
On the right of this drawing is called a switch loop. Those are wired with black-white cable out of sheer practicality... the white is not neutral, but is switched-hot. Legally the wire must be marked, typically with a few wraps of electrical tape. Black will suffice, use red if you have it.
This switch loop arrangement is actually obsolete. In new work, code now requires neutral be brought on switch loops. In that case, all 3 wires would be extended to the switch, and neutral would not be used until the happy day you get a smart switch or motion sensor.
The color of the wire-nuts do not matter, as long as they fit the wires - they have a lot of range. Yellow is best suited for all the connections here.
All these drawings omit ground wires. That's commonly done on illustrations like this. Nonetheless, all the grounds do need to be there and be hooked up.
One last thing: You'll get way less light. A dual 4-foot fluorescent (62-90 watts) produces the same light as 400-500 watts of tungsten bulbs. If you hate the fluorescent flicker, lousy cold-start or hum, or the ballast or starter has failed - electronic ballasts fix all that! And you can retrofit your existing fixtures. If you hate the fluorescent color, they fixed that too - tubes and LEDs now come in excellent color rendering (80-90 CRI) in 3000, 3500, 4000 or 5000k color.
Direct wire. Worst case you have to change a lampholder.
Plug-n-play doesn't allow a ballast, it requires a ballast! And of a specific type - if the fixture has the wrong type, you would have the ignominious task of buying a new ballast and also possibly change lampholders too. It's one more thing to maintain.
Best Answer
Q: Is this possible? How could I do it?
A: Certainly possible, but there are some obvious issues:
Have you considered an LED lookalike for the T8 fluorescent? There are units available that run directly on 120v line voltage. The initial lamp cost on these is high, but if you are building a custom piece of light fixture art, that may not be an issue. One example is model 1920WH-L (I am not endorsing this particular unit):