Suzuki demands that you have a piano. A keyboard just doesn’t have the touch of a piano—even thought there are ones these days that advertise they have a very similar feel. You need to have a keyboard that is force sensitive. It can feel how hard you hit the key, and produce an attack and the volume that you’d get from a piano. I don’t think they can really do it.
With piano, it really matters how you stroke the keys. You can press them straight down, and you get one kind of attack. You can stroke them as if petting an animal, and get another kind of attack. These greatly affect the richness of the not you’re playing.
The smallest real synthesizers (as in not kids) must have at least four octaves. But if you do get a synthesizer, get a full keyboard if you can.
However, I think it is far preferable to get a piano. As @MindErrantry said, check your local Craigslist or just wander around the neighborhood or check out local meeting halls and schools. There are dusty old pianos that haven’t been used in years everywhere. Get it home, get it tuned (tuning is extremely important) and fixed up, and you can have a piano basically for the cost of transporting it. We got our piano for free, and just had to pay for it being trucked 200 miles (it was a good piano). That came to around $500.
If you have three strong friends, you could move it yourself, but you have to be really careful with it.
Synthesizers are fun, and many people only play them. If you just want to fool around and make a few songs or accompany yourself as you sing some rock or popular tunes, a synthesizer should be fine. It has all kinds of sounds, and you can mess with them and have a lot of fun.
If you really want to understand your instrument and gain enough skill to do interesting things, you should take lessons. You’ll learn how to read music as well as learning good technique.
We started with a synthesizer, but that was mainly for me. I’m not a piano player. When my kids started lessons, we were told to get a piano, and now the synthesizer is mouldering away in my parent’s house. It gets taken out for the kids to practice on when they go up there for a visit.
Piano teaches you theory, just by looking at the keys. I’m a trumpet player, and theory never made any sense to me. Having a piano in the house has taught me so much about theory. I now know what Jeruba was talking about with her chord sequences. You can see a seventh on a piano. On a trumpet, you can only think it. That kind of theoretical approach is just too hard for me.
I like to play by ear, but that’s because it takes me away from myself when I do that. It’s an altered state of consciousness. I can read music, and sight read just fine. But playing by ear—well, I guess you can’t improvise if you can’t do that. Improvisation is the best! (and that’s a metaphor for life, too)