I've been playing both piano and drumset/percussion since I was a kid. I've also dabbled in guitar, bass guitar, and sax as well as MPC's and turntable scratching. Eventually I wanna go back and master than sax, get my scratching better, and learn the cajon.
I'd honestly say pick the instrument that intrigues you the most. There's a cheap student version of almost every instrument out there you could practice on until it becomes second nature and you decide to buy top of the line gear. Becoming proficient is an uphill battle that'll take at least a year so you need to pick the instrument you think is the coolest. Otherwise interest just tapers off and you give up.
Learn how to read music if you can. It makes learning songs a million times easier.
There are many suggestions for the piano. Can I learn it on a keyboard or similar? Getting an actual piano is probably out of the question.
You can learn on a keyboard. The feel can be way different but it's good for getting started. I bought my girl a $300 keyboard when she wanted to learn the piano. You could probably find a used upright piano for around $400 but it wouldn't be portable and you'd have to worry about tuning it.
The guitar is intriguing. This is very different from the bass guitar my friend wants me to try, right? They look the same, but I think they are played differently. Pros and cons?
Guitar = 6 strings. You start off learning chords (multiple notes played at once
Bass guitar = 4 strings that are longer to make deeper sounds. You almost never play chords on a bass so you start off just plucking strings with two fingers. Bass guitar is easier to get started with.
Drums are some of the most fun IMO. You can get an electronic drumset for well under $500, learn some basic technique, and be on your way.
I didn't know renting was even an option. Maybe the cost requirement was just diminished.
If you could get a rental with actual in-person instruction, that would be your best option and you'd probably learn faster that way too.