I am also 14. I started my business when I was 12. As of right now I have around 15 customers I mow on a regular basis, and 5 more that sometimes have me do their yards. I started off with nothing but an old lawn boy push mower (14 yrs old) and $400 in the bank. Now (2 yrs later) I personally own TWO zero turn lawn mowers, one is a 2005 Exmark Lazer Z with a 27hp Daihatsu Diesel and 60" Ultra Cut deck (anybody have any reviews to give me on that mower?), and other is a New Holland mz16h, which is NOT my first choice of a mower and I bought the Exmark to replace it because I was sick of it. But it's good for smaller yards and works fine as backup also. I also own a 21 inch Honda Self Propelled push mower, $400 new. It's a good mower for what I use it for, I've never had a problem with it. I use it for back yards where I can't get my zero turn or tight places, sometimes around streams, etc. As a backup for it I still have the 14 yr old Lawn Boy, which runs just fine. I own a Stihl KM 130 and have the weed eater, edger, and hedge trimmer attachments for it. I also have a Stihl Br 600 magnum leaf blower, for cleaning up after a mowing job or blowing leaves during the fall. Then for the winter time I own two snow blowers, both two stage 24 inch blowers, one is a practically new Ariens, and the other is a backup Toro.
Do I need all that stuff? Probably not. But that goes to show that it is very possible to be relatively successful at a young age. At this point all that is holding me back is my schedule, meaning school and what not, but also that because I'm 14 I can't get a truck and trailer and go get some more customers, I have to stick to my neighborhood. I just drive my lawn mower with a trailer behind it to the job. So I am pretty much stuck in a 1/2 mile (or less) radius, which I've done pretty well with, but I'm never going to get rich with just that.
In your case I would recommend sticking with your push mower until you have at least 5 yards. Also I would strongly recommend getting a weed eater and a leaf blower. They don't need to be fancy, they just need to work. I think you could get a decent used trimmer for $100 and a decent used handheld blower for $100. It isn't the best option, but you can use the trimmer for edging around the sidewalks and driveway if necessary, and trimming everything up plus blowing off the grass from the driveway and sidewalks goes a long ways to satisfy your customers. By the time you have 5 customers or so, if you think you can still expand, and you are making a decent bit of money, at least enough to justify spending more money on your company, I'd buy either a regular riding tractor, or you can probably find a good commercial grade walk behind for about $1500-$2500 that will be better than a regular riding tractor for sure, and will get you to where you can pretty easily do at LEAST 12 lawns. Some guys actually prefer walk behinds to ride on zero turns. Now don't forget, I'm 14 also and I have an abnormally busy schedule, between school and a number of other major commitments. So I only have about 1.5-2 day worth of time during the school year to work, and with the equipment I have I can complete all my work in that amount of time. If I didn't have the equipment I have then I'd not be where I am now.
Good luck, I always am happy to see other kids my age out there starting a business.
Also, forgot to mention. As far as insurance goes, don't worry about it at this point. I don't have any contracts, it's all cash and stuff you just shake hands on. I do plan to start paying taxes this year though, which I haven't done in the past. My plans are to get insurance if I expand to the point that I have a truck and trailer and am driving around running a full fledged business. At this point though in my area at least it is not necessary. Most of the lawn care companies around here don't even pay taxes, forget insurance. I just wouldn't worry about that at your age. Later on though I think it is unwise not to, if you get 20 customers that you are mowing for every week than maybe it'd be worth getting insurance through your parents or something, and you are always at risk without it, but if you don't do anything dumb you will be fine.