The first thing you'll want is a horn that is decent quality and in good condition. By "decent quality" I don't mean a pro quality horn, I mean something that is well built enough that things are aligned properly and don't break or constantly go out of adjustment. Unfortunately, there are a lot of junk horns directed at the student market online and in some music stores. It's not easy for someone who does not know saxophones to spot their flaws until they realize that pads don't seal, keys start sticking, solder joints come undone, and parts literally fall off. If you're in the market for a student horn, your best option is to buy a used horn made under an established brand. Because student horns are constantly going back on the market as students tend to either quit or upgrade their horns after a few years, it's not hard to find reasonable deals.
The best known student horns are the Yamaha 23, Selmer Bundy, Vito, and Conn Director (shooting star). The Yamaha 23 is famously reliable and easy to play. The Bundy I is a solid, American made horn from the 1960s. It's a variation of the Buescher Aristocrats that were produced in the 1950s as student horns before the Buescher company was bought by Selmer. Aristocrats/Bundy Is were the first horns for practically a generation of players. The Bundy II is a more modern design, but not as highly regarded as the Bundy I. Vito was a long-running line of student horns that were solid until their production was farmed out to Asia and their quality went downhill. The Conn Director was a reasonably good horn from the late 1950s through the 1960s, when they were produced in American factories. If you get one, you want one with "Elkhart, Indiana" or "USA" engraved on it. The ones produced in Mexico from 1970 on were of unacceptable quality.
The best thing about old Bundys/Aristocrats, Vitos, and Directors is that they are dirt cheap on the market - under $500 for a tenor in top condition. Avoid getting one that needs major work beyond simple pad, felt, and cork work. It is easy to exceed the value of an inexpensive horn in repairs. Used Yamaha 23s cost more.
Jupiter is a relative newcomer from China that produces a horn similar to the Yamaha 23. Their newer horns have a good reputation. Their older horns (up to about 2000) are known for quality problems.
The most important thing for any developing player is disciplined practice that results in good playing habits. If your budget and schedule allow it, in-person lessons are worthwhile. If you have the discipline to be self-taught, there are online lessons (free and paid) linked through sites such as http://www.bestsaxophonewebsiteever.com/