The basic tool for finding lost arrows is a hook made out welding rod or some other stiff metal rod that is attached to a wooden handle or golf club handle, etc. if you know within a few yards where the arrow went in, this works fine for raking the arrow out of the grass or weeds.
If you just have a general idea, a metal detector is good because you can cover more ground. They don't work very well unless they are close to the ground, so if you have more than 6 or 8 inches of weeds growing up, it probably won't work. The more metal, like aluminum arrows, the better they work. A carbon arrow with an aluminum insert and a 100 grain tip is going to be harder to find. Wet ground is difficult.
Mine is a Discovery 3300 that I got from Amazon. It isn't the cheapest, and a long way from being the most expensive. Amazon only had about a hundred different models to choose from, and I have no idea which one is the best for finding arrows. I doubt if it will ever pay for itself, but it is so frustrating to lose an arrow that I don't care.