Not a bad idea, although if I were to play Devil's Advocate I would say once you have expended 6 months worth of rental fee on a rental horn, you will likely have paid as much as what it would cost you to buy a horn to begin with. And the thing is...if you decide sax isn't for you....you'll never get that rental $ back, while you can at least resell your own horn and recoup some $.
All replies above make good points. Teacher: mandatory. Good-working horn: mandatory. Easy to initially learn: yes, sorta. Hard to master: indubitably. Fun: hell, yeah.
Now...do you have any idea WHICH sax you wish to play (Alto, Tenor, Soprano, Baritone) ? A lot of folks answer "oh, I dunno". If that is your case, listen to some music and find the sax sounds you like....and figure out which horn it is.
Many folks choose Alto because they have the (mistaken IMHO) impression that it's 'easier' than the others or more appropriate for young players. In reality, I think the Alto gets that reputation simply because it tends to be the cheapest of all horns to acquire.
I always suggest this to a brand new player:
1) Make a preliminary decision on which horn...you can change that decision later.
2) Go buy a decent but cheap mouthpiece and 2 or 3 reeds. Like a Brilhart Ebolin 3 or Yamaha 4C mouthpiece.
3) Go online and find free vids for "how to blow into a saxophone mouthpiece". Learn how to do it.
4) Go to a local shop and ask to play a few horns (no point in asking for their pro models, but no need to really pay attention to their price tags either...because you are testing yourself, not the horn). If they have both new and vintage horns, try both.
5) Take it to a practice or tryout room and slap on the m'piece and just see if you can get the horn to speak a few notes. To do this, you may wanna also bone up on how to finger a few notes on the instrument (see free vids for this, also).
6) See if you like it. Does blowing into it and making a sound 'move' you positively ? Or does it make you feel like "oh...this sounds nothing like what I wanted it to" ? If the latter, then repeat the process trying another size horn (i.e. if you did this with an Alto and you didn't feel good about it, try it next time with a Tenor).
Glad to have you on the Forum !!!