"come a long way" usually means that someone has IMPROVED a lot. It's almost always used in a perfect tense (has come a long way OR have come a long way) EXAMPLES: My brother used to get very bad grades in math, but now he is the top student. He has come a long way. I used to hate eating broccoli, but now I eat it every day. I've come a long way! John used to be very fat, but he's lost a lot of weight in the past few months. He looks great! He's come a long way! "Go a long way" means that something is more than enough and will be useful. (it's less common to use than "come a long way") EXAMPLES: This food will go a long way to help the poor. (This means the food will help the poor people a LOT. It will be very useful for them, and it isn't a small amount of food.) This new, more advanced lightbulb will go a long way in increasing energy efficiency. (This means that the new, better lightbulb will be very useful because it will increase energy efficiency a lot) Just a little time and effort every day can go a long way toward improving your English. (This means that a small amount of time every day will be very useful for improving your English) Sometimes, "come a long way" and "go a long way" can be used similarly to talk about improving, but "come a long way" is ALWAYS used to show a CHANGE and how something is different than before. "Go a long way" emphasizes that something is USEFUL, usually how a SMALL amount of something can be VERY USEFUL (small amount, large use). I hope that answers your question!

The difference between "come" and "go" is related to where the "observer" is. We use "come" when the person is travelling to us, and we use "go" when the person is travelling away from us. For example: I am in Berlin. My mom is in New York. I would say: "My mom would have to come a long way to visit me." I could also say: "I would have to go a long way to visit my mom." (since the direction of travel is away from my current location)