Socks sliding down toward your ankle are bad enough. When they slip all the way into your shoe, it can be uncomfortable as well as unsightly. It's a common-enough problem that several solutions have been devised -- or, in some cases, improvised.
Sock garters are old-fashioned and sometimes a bit expensive but highly effective. They consist of a flat elasticized band that loops around your calf, plus clips to attach to your socks. They come with either one clip in front or two clips, one in front and in back. You can find sock garters at high-end menswear stores and vintage stores as well as online.
If you don't want to invest in a pair of sock garters, you can try putting an elastic hair band around the top of your sock instead. It will act as a makeshift garter to keep your socks aloft. This solutions is more prone to slippage than a true garter but it's also easier to improvise. You will need to find a band that can stretch far enough to accommodate your calf (without cutting off your circulation).
If all else fails, you can invest in a product called "sock glue." It's exactly as advertised -- a skin and fabric-friendly glue that washes off at the end of the day, but in the meantime sticks your socks into place against your calf. Sock glue can be hard to find -- your best bet may be to order online. Irish dance suppliers often carry this product, as do Japanese fashion sites.