Well..I used to play violin, my first instrument in fact, played for about ten years. My violin playing career ended with being kicked out of the school orchestra for going to see Status Quo rather than playing in the school operetta (HMS Pinafore if i recall correctly...as if could ever forget!)
Still got a few knocking around, pick it up and have a scratch every now and then.
When i started playing guitar (bass actually, mid-teens) it literally took me years to get comfortable with the strings on the bass being opposite to those on the violin (violin goes G D A E low to high)
Picked up sax a few years later (via the flute) played it for about 30 years now.
So..Sax...I'd suggest a Yamaha YTS-32 tenor, great horn for a starter/intermediate, well made, in tune etc etc. Probably around $1000 for a decent example. Match it with a Yamaha 4c mouthpiece, your bog standard cheap, well made starter mouthpiece, and you'll be good to go.
Of course, there's lots (and lots) of other options around, especially these days with the no-name brands coming from China/Taiwan/ebay. Some of these horns can be good, i actually have a cheap Chinese curved soprano that works fine, but it's a bit of a crap shoot.
"Jupiter" is another known brand (at least here in Australia) that's not too bad...thing about saxophones is there's a lot of mechanical bits and airtight seals and shims etc that need to be working properly and "in concert" to get a decent sound. A poorly functioning horn is a whole world of frustration...especially if you're not sure if it's you or the instrument itself.
A lot of beginners are discouraged by bad horns.
And once you get the hang of it all, you can start delving into the wonderful world of mouthpiece designs/options/opinions..
The "Sax on the Web" forum is a great resource, plenty of advice there...
Also check the book "The Art of Playing Saxophone" by Larry Teal, a great treatise on the whole business.
You'll also be diving into the whole world of breath control and embouchure...very different experience to playing guitar.
Violin...I have no idea what violins are worth these days, but as an acoustic instrument you're looking for the same kinda things as you would an acoustic guitar, decent build, nice tone, etc etc. I'll take a punt and say the cheap Chinese ones are probably a safer bet than the sax's, just because there are no real moving parts.
Intonation is the big deal here, left hand..where are those notes anyways?
The other big deal is bowing articulation, right hand, which takes quite some training to achieve.
The mastery of these two skill will torture your close family/neighbours/anyone within range for years to come!
Actually, intonation is a big deal on sax as well.
And constant practise to maintain your embouchure, as it deteriorates quite quickly and takes a few days to build up again.
Like all wind instruments, every note needs to be coaxed or "produced", pushing the buttons/keys is just the start of the process.
And then there's the altissimo...
Conceptually, it would be an interesting exercise to learn the sax as a completely new physical skill-set to master.
OR...get a cheap solid body electric violin, plug it in to your Axe FX (with emphasis on the "fx"), turn it up, and..GO!
Sounds fun either way to me!