Ballet classes at age 3 are not going to be much more than a 'be an acorn growing into a tree' level, there will be no formal classical ballet training yet. Same with piano, a 3 year old's hands will not stretch across the keys.
By all means start her in a ballet class but accept that it will be basically just movement to music until she's a bit older.
I wouldn't start with piano just yet as she will be in the key banging stage. (I started learning the cello at age 8 & the piano at age 10)
No offence but you sound a bit keen for her to start formal classes, she's only 3. She may not like them & may not be able to sit still for them yet. Calm down a bit & just dance to Led Zeppelin in your living room sometimes & play classical music in the background frequently/at meal times
My wife and I had a Yamaha organ, and our DS was 'playing' on it from age 2. By 4 or 5 he was playing tunes, starting to read simple music, enjoyed creating different sounds, and the automatic rhythms gave him a good sense of 'time'.
A Keyboard, with full size keys, at least 61, and ideally 76 if you can afford it, can lay a foundation of an awareness of how sounds and rhythms are created, and how notes can make simple tunes. If you have sufficient experience to guide her through the basics tones and beats, she could have a head-start for more formal music in a couple of years time. BUT it should always be 'fun' and learning should be at pace she is happy with.