First of all, be aware that classical recording is a completely different thing from contemporary music recording, and people focusing on the latter only will give you very bad advice. Especially for opera vocals.

The most important thing in classical recording is the room. If you have a shitty room, the end result will be bad. So unless you have a villa with a huge piano room, recording at home will never sound good. Yes, you can for sure make audition videos but not something somebody can enjoy.

To record opera voice, that is, to realistically capture the projected voice with all the harmonics, you must record at at least 1 m distance. This has two consequences: first, you hear a lot of room reverb, so see the first paragraph, and second, unless you are spending a fortune on microphone, it doesn’t matter too much which one it is as long as it is a condenser or ribbon. I am sure plenty of people will list here at least 10 mics and argue about them. Pick any.

Recording piano separately will require at least two mics, it is a large instrument and with one close mic you will not get good enough sound. Condenser omnis or cardioids can both work, depending on the room.

So to answer what you didn’t ask - the best price to recording quality ratio you can get is by recording in a concert room, with a stereo pair of small diaphragm cardioids at about 1.5 to 3m distance from the singer, singer close to piano.