Well, it is funny that you should ask. I did it all myself and it worked out for the most part, but I will share my experiences about what NOT to do!
First of all, booking airfare is easy and I use Hopper to book airfare. This app is great, because you can put in the times and where you want to go and it will tell you what current fares are, if they are likely to go up or not and when is a good time to purchase. So, that is how to handle airfare to-from Europe.
I would recommend planning your trip, so that you fly into City A, and fly out of City B, where you plan to end your trip. In our case, we started in Munich, then took a train to Prague, train to Berlin, train to Amsterdam and I had planned a final train back to Munich, but it would have been 10-14 hours by train and in the end, I spent $300 for two one-way plane tix and it was a 1:20 flt. I would have booked us into Munich and out of Amsterdam, but I bought the plane tickets before I knew exactly where we were going.
Within Europe, you can travel by train, which is easy and relatively affordable. I spent about $800 on a 15-day twin German rail pass, for two of us, that allowed us to travel on any trains in Germany at any time, within that 15 day window. It also covered domestic trains in the cities, BUT NOT SUBWAYS! I also made a bad assumption that it covered trains from Germany to adjacent countries, on German trains! Not so! I ended up paying a bit more, for the train leg from the border of Germany and Czech, to Prague, and from the border of Germany and NL to Amsterdam. I probably would have been better off with a Eurail pass, which costs a bit more, but covers trains to all countries.
Now, I would also consider the cost of flying between countries within Europe also, because this is faster and may surprise you at the cost. Trains are convenient and inexpensive. Recommended for max 4-5 hour trips between cities. When you factor in the time to get to the airport, security, the flight itself and then from the airport to the city, often a 4 hour train ride is just as fast as a 1 hour flight! Trains are easy to hop on and off, no security and airport delays.
As for accommodations: AIRBNB! It is THAT simple! This is the BEST WAY TO GO! Look at ratings and see what is there, but you will save money and we had GREAT experiences with this! In Prague, we had an entire flat to ourselves for about $75/night! In no case did we pay over $100/night for Airbnb's, while we paid over $150/night at average hotels and they were not as comfortable.
So, you could use a travel agent, but I really don't think it is necessary. It really depends on who you want to travel. This was my 16 year old son and myself. We had a backpack each, med size and no checked baggage. We wanted to be flexible and free to change our minds about where we wanted to go. The rail passes made sense in this regard.
Once you arrive in a city, public transport is very easy and inexpensive in most cities. Worst case there is also uber, for a bit more, if you in a pinch or don't want to deal with subway or train.
So to sum up:
Pack as lightly as possible, so you are not lugging a bunch of luggage around.
Use Hopper or other site to find the best flight rates and times.
Decide on rail or planes within Europe. We had a good experience with trains, just get the right kind of pass.
Airbnb is the cheapest and most comfortable accommodation option in my opinion!
Public transport within Europe is cheap, fast and easy.
Use a credit card WITHOUT international transaction fees as this will give you the best exchange rate for all transactions. You will need minimal cash in Euros, etc. Cash exchange is NOT going to be at the best possible rate.
I highly recommend Prague and Amsterdam! Munich was preferred by us for the German experience. There is also France, Italy, Greece of course. I am just sharing our experiences and I hope that it helps.