Re: Drummer picking up the Saxophone... Need advice!
If I recall, Coltrane started out on Alto.
I did also and played for a few years before adding Tenor (and mostly playing that now).
It is generally easier to go from a smaller to larger horn, but if you know you want to play tenor it probably would be a good idea to start looking. I wouldn't get in any huge hurry, play as much as you can now- everything you learn on the alto will work on the tenor, with the exception of your embrochure will be different.
You don't *need* a teacher but it may help you.
Some tips for self learning that may help.
Breath from your diaphragm.
Record yourself and listen. Keep those and listen again in 6 months. Hear you what you like and what you don't like.
Listen to players who you like their sound.
Play lots of long tones, concentrating on sound.
Breath from your diaphragm!
Breathing and breath support is hugely important in your sound and ability to keep in tune.
On those long notes, play against a solid tone and match it. Make sure when you attack the note (either softly or hard) it is right on pitch right away and stays there (later you can mess about it bending it when you want, but start out right on).
I can always hear players with not a lot of time, most all have poor support and a 'whimpy' sound.
Even if you like a light sound (think Paul Desmond) his pitch was right on and stayed solid.
Those are just a few off the top of my head.