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Don't install Ubuntu, it's not very lightweight I don't think. I have installed Linux Mint on a few relatively weak PCs that couldn't run Windows comfortably and Linux Mint runs wonderfully. It's also very easy to use if you're used to Windows: My girlfriend and his father neither have any trouble with it, and neither of them (especially her father) is very tech-oriented. For those specs, staying with Windows is just not an option. There's even more lightweight options than Linux Mint, but they're generally very ugly and Mint is pretty lightweight already.
There's two main options (and a few others too) for the desktop environment in Linux Mint: Cinnamon and MATE. Some say MATE is faster, but I haven't noticed any difference personally and a few tests I've read indicated there's not much difference. Personally I'd pick Cinnamon because it runs about the same, looks a lot better, and you can change the desktop environment after installation if you want/need to (it's really easy, just install it from the store-like thingy and the next time you log in, choose the new desktop environment).
EDIT: The computers I've installed Mint on have 2-4 GB of RAM, but I think it should run with even less. The official minimum is 512 MB, with 1 GB recommended for comfortable use.
