I spend 4.5 months travelling this summer, so hopefully I can give you some up-to-date info.
I should mention that I have never been to Africa, so I have no idea about anything on that front. I have been to a vast majority of European countries though, so this is what my input will be for.
First off, i'd cater countries and cities based off of what you want to do/see. Are you interested in another culture in particular? Are you looking for automotive related things to do? Are you a big foodie? Do you have a maximum of a $2000 budget to spend during those 10 days? What sort of activities do you want to partake in while you are in Europe? etc. I think answering those types of questions will certainly help you narrow things down.
Secondly, if all you have is 10 days, I would strongly recommend visiting fewer cities/countries and doing more, rather than visiting as many places as you possibly can. You can always go back to cities/countries that you don't get to, but if you rush through some now, you'll probably find yourself feeling like you shouldve spent more time in the places that you visited.
Now... Starting in Amsterdam - I would highly suggest at least a day or two there. The city is beautiful, and it's truly something to experience. Depending on the kind of person you are, you may want to stay longer. From there you have far too many choices. You can take a bus, train, car, or fly to places. I'd narrow down a certain region you want to visit early on so that you can decide what to do. For simplicity sake, I am going to recommend that you stay in this area of Europe for the trip because it will likely be easiest and you'll get to see a fair bit.
From Amsterdam, I would recommend visiting Bruges, Belgium for sure. You could take a bus or train to get there, but I rented a car when I was there and I say that it was definitely worth it. It gave me the freedom to do whatever I wanted, whenever I wanted (including overnight drives to other countries). You can check this city off your list in probably half a day (which is why it would be nice to have a car). From Bruges, I made my way to Dunkirk, France this summer and I thought that was a really neat city. You have some pretty cool beaches with great history and architecture. If you want to go the other way, i'd head towards the city of Liege, Belgium. I stayed there during the Belgian Grand Prix, and I loved the city. A day here would suffice. From there you could make a pit stop at Circuit du Spa-Francorchamps if you're into that. Maybe there's an event going on here? From there, you can drive down towards Luxembourg City, Luxembourg. This is a pretty neat place, but again a day here would suffice. After that, I would simply do a loop of Germany. Stuttgart, Munich, Frankfurt, Cologne, each of these cities has a bunch to offer, and if youre a motorist, many museums as well.
There's so much I could write about so many cities, but without having a general idea of if you want to do Scandinavia more than Germany/Belgium/France, it's hard to pinpoint exactly. Hopefully this helps you though! If you have any questions, certainly let me know!