I mainly grab free art opportunitys. It's the best way if you can't afford or simply don't want to throw all your money out for art, which is absolutely fine. Not everyone wants to spend money on art.
But the most important thing: don't beg.
Search for some open request posts on the internet. Deviantart, here, Instagram. Whatever. Do not randomly message artists and ask for free stuff (unless they're clearly open and even want you to message them if you have a request) that's a little rude in my opinion. Especially when they're obviously doing commissions.
Not being able to commission your favourite artist sucks. Speaking from own experiences but if you work hard and save up it's definitely possible. I was also able to buy quite a few pieces because of simply hard work and patience. Crying over it won't do much in a long run ^^
And.. for the love of god don't do fanart for a popular artist because you expect something in return... Please... No.. that's wrong and rude on so many levels