Most people start on alto because most people start when they are children. So, a big part of the choice of alto vs. tenor or baritone is the size and weight.
I agree with this, and we can add 'price' too. Altos are the cheapest saxophones, relatively speaking. Therefore the biggest sellers....
But indeed if one is a late-teen or older, the 'start on alto' advice becomes sorta moot. Pick the horn with the sound you like best. For some, it's Alto, others Tenor, others Baritone.
As noted by others here, stay away from uber-cheap, new saxes. I am talking Altos for $400 or below, Tenors for $600 or below.
They will be quite bad, probably from the get-go....and you may well find they do not last more than a couple of years without requiring significant repair/upkeep.
So....the cheapest NEW Alto of any repute is going to run at least $750, and that is for a store-brand horn (but again, one with some repute from a store of some repute).
A big-name new Alto is gonna be at least $1400.
A used Alto can be had for as little as $300 in playing shape - but buying used requires some care and thought. As a newbie, you want the seller to guarantee it to be in good playing shape - or they will take a full return.
Tenors, the cheapest NEW ones of some repute gonna be $850-900; a new big-name student make gonna run almost $1800; a good, serviced used one can be had for around $500.
I am not necessarily gonna bandwagon the Yamaha 21/23's.
Yes, a solid suggestion, I sell tons of 'em, and they are ubiquitous - but there are a number of others equally as good and priced similarly, used: Jupiter, Buffet, some Selmer USA's, etc. in contemporary horns....then the good older models such as Kings or Conns, etc.....
The good choices abound, really.
If you wanna jump on something safe without doing a lotta research, I just finished a YAS-23 Vito branded Alto which I am selling for under $300....it's in my listings section. (Apologies for gratuitous plug)
Hope this helps. (Saxinette....yeah, wow...a new one on me !!!!!)