Remote Desktop Support

I have an elderly mother who owns a Mac computer and lives in another city. I'd like to help her with computer related tasks by viewing and, ideally, controlling her computer screen remotely (over the Internet).

For every non-Apple computer system I've every used, this has been a trivial task, but I've been unable to do it under MacOS. The obvious answer is to switch to Windows or Linux, which I, and everyone I know, is indeed planning on doing. However, it's not a great option in the near term for her, since she's in her 90's and has some memory problems, so probably won't learn a new OS that fast.

I'm aware that Apple sells a product called Apple Remote Desktop for $79.99. However, from what I gather, it's too unreliable and buggy to use.

Can someone suggest a software program, or programs, I can use for remote access to my mother's computer? Ideally, it will not have monthly charges, and it would be cross-platform, so we can all continue using it after discarding Apple technology.