I believe you very much can earn enough to take lessons, and it might not take as long as you think! Lessons are easy to afford when compared with the cost of buying and owning a horse for the first time.
As far as the basics (beyond what most people would call raw beginner), I think it takes about 3-6 months of once a week lessons to get to a good, serviceable level of knowledge. You would not be an expert or an advanced rider by any means, but to generally know how to ride at all 3 gaits comfortably, groom and tack up, basic horse care, and how to handle common riding problems as they arise and handle horses comfortably.
Everyone is different, and you might progress faster or slower compared to someone else. The key to this is finding a good instructor you are comfortable with, and who has teaching methods that you work well with. This person knows how to encourage you without babying you, and how to give constructive criticism without making you feel like you're three inches tall. They also know how to challenge you when you need to be challenged so you gain confidence.
As far as how much money you need to save up, it pays to shop around, and it will depend on where you are in the country and what discipline you want to pursue. If you just want to learn enough to ride and go on trail rides, you'll pay less than someone who wants to show seriously. Start thinking about these things:
-What type of teachers do I get along best with?
- What sort of riding do I want to pursue? Do I just want to ride and be around horses, or do I want to be competitive? (If you don't know, just go with learning the basics- you can always change your mind later on).
- Look at the different disciplines (Western, English) and see what speaks to you. You can always learn both, but I'd recommend starting English because if you decide to go Western later, the transition is much easier than going from Western to English.
- How much is a reasonable price to pay for the type of instruction I need? Do some research and look at trainers that offer the services you are looking for. Every trainer is different. You might pay $25 for a lesson someplace, then someone else might charge $55.
Start talking to horse people now. Go to tack stores, feed stores, horse shows, anywhere horse people gather, and talk to people about learning to ride. They can recommend someone, or you might meet a trainer you really love and want to work with. Some farms will even let you work to partially pay off your lessons, but not every barn offers this.
Go for individual lessons first. Group lessons are difficult when you are a raw beginner, even if they do save money. You will learn and progress much faster when you have the instructor's whole attention- you can go for group lessons later when you have the basics down.
Good luck!