It is far better to start with an 88 key piano, as the quality is far better, and you don't have to reinvest later once the notes start leaving that range. In the long run it will end up saving you money, which is why I suggest you wait and save up. Another option is to buy an upright piano used, some of them can go for as cheap as 100 dollars, and still be in relatively good condition. Kijiji is usually very good for this. 88 keys are better because of as was mentioned before the touch sensitivity, as well as the better overall sound, and usually more options, though sometimes a 61 may have the same amount of options. Also as was mentioned the lack of weight on these makes it almost impossible for you to play on a real piano or even a regular digital piano. 61 keys often only have (basically) the sustain pedal since if they do have an input for the mute pedal it is so poor that it may as well not even be there.
Your teacher will probably start with some basic basic theory and repertoire, and then advance you from there based on your talent and dedication. Some better ones will also make you study the history and teach you how to analyse a piece of music properly, and thereby interpret the music as it was either meant to be played, or how to create your own rendition of the music once you start getting into grade 3 or higher level playing.
If you have a good teacher they will ask you to get at least an 88 key digital, while some will require an upright. In this it is better to ask your teacher as everybody has their preference and it is better not to try and guess your teachers intentions, and then have to spend more money. If you are serious, then buying a 61 key is almost guaranteed to be a waste of money though.
My lessons typically go with reviewing what happened last week, then assigning the scale to study for the week, then we move on to repertoire, since I am far beyond learning new theory until I attend university. (I'm still not sure what if any I will learn there). Your teacher will guaranteed start by teaching you how to read notes and other basic theory. If you want to get a headstart on that you can visit www.musictheory.com
Goodluck and I wish the best to you in learning the greatest instrument, well, ever :)