Travel to London, Paris, Barcelona

This is one of those questions that just begs for more information because our only answer is "maybe." On this site most posters will do their own travel arrangements. Most would suggest that doing it on your own is always cheaper because organized tour operators have to make a little money somewhere. If you can find a company that does exactly what you want then you have a basis for comparison. A tour company takes a lot of uncertainty out of the planning. And, if this your first trip to Europe and traveling by yourself you may want to pay for the convenience of an organized tour. Only you can value what is important to you. But since you have time, hit your local library and check out travel DVDs and guidebooks for those three cities. See what appeals to you. Then come back and we may be able to help you with your detail planning.

Ten days only really gives you a good eight nights on the ground. I would strongly suggest that you only do London and Paris. London gives you a start when they do speak a somewhat recognizable form of American English and then train to Paris. Use an open jaw - multi city - ticket into London and home from Paris.

All three cities are well covered by guidebooks (I guess I don't see any reason for packaged tours or travel agents in this case because there is so much info already out there to help you plan). Ten days is not enough time for such big cities though, I would consider cutting one and planning the hotels and attractions on your own. There will be so much to do in each city, I'm not sure you'll have enough time for excursions (if by that you mean side trips out of the city).

Yes, 10 days is not enough for all 3 cities, both London and Paris need 5 days. Are you including travel days in your dates, you really only should count days at the destination. You could do 1 day trip from each city: Windsor and Versailles - compare and contrast 2 very different royal palaces.

You could do all 3 in 10 days, but it would be rushed and you would waste time travelling.

London+Paris is easy because it is only 2½ hours from one to the other by train, city centre to city centre.
For info on the trains look here: https://www.seat61.com/London-to-Paris-by-train.htm
The company running the trains is "Eurostar". Times and buy tickets from their website here: https://www.eurostar.com

Fly "open jaw", that is into London and back from Paris (or vice-versa) that avoids wasting time and money doubling back.

Edit: You have somewhat of a learning curve before you. You need to read up how each city's transport system works. Tube +buses in London and métro + buses in Paris. Both systems will get you where you want to go relatively cheaply, and a lot cheaper and more flexible than a tourist excursion.

If you want to see all three of these cities in 10 days, and you are a first time traveler to Europe, you definitely should take an escorted tour. Such a tour can move much faster than you can on your own, and you won't have to use time figuring out each city (there is definitely a learning curve in a new place, and it is even more pronounced when you haven't been to so many places before).

If you can cut back your plans and see only two cities, you can do this on your own, if you don't mind (or, like many of us, actually enjoy) lots of research and learning. These are large cities with lots to see and do; some things in all of them will require advance reservations (particularly Barcelona); you'll have to learn about local transit; etc.

Everyone is different. For some, they love just showing up and letting the tour company take care of the rest; for them, dealing with planning and details is not a vacation, and making decisions is best left to others. For instance, a coworker says that when at home, she and her husband are always going "Where do you want to eat?" "I don't know, where to YOU want to eat?" over and over. Their whole trip would end up this way if they didn't go on tours. She also has a serious case of FOMO (fear of missing out); going on a tour assures her she's seen "the highlights," which is very important to her. For instance, she's one of those people who feels that a first trip to Italy is not complete without seeing the Leaning Tower of Pisa, and that when in Paris you have to go up the Eiffel Tower - regardless of whether she ends up enjoying these things. Taking an escorted tour assures her these boxes are checked.

Others like having the control to go when and where they want, rather than being led around at a fast pace; for them, deciding exactly what they want to do is fun rather than stressful. They don't feel the need to go places just because "everyone" says you have to, and aren't worried if they "miss" a place, because they used their time to see something else instead.

There is no right or wrong way.

If you do want to plan and book things yourself, start by getting Rick Steves Europe Through The Back Door. You do not need to get most recent edition; a library copy or used copy that is a few years old is fine. This book will give you the nuts and bolts of European travel (what to expect in a hotel, how to handle money, how to order food, etc). You'll start to see what's involved in planning it yourself, and you can see if it's the right way for you, or if you prefer a tour.

If you do decide to go on your own, you can then get Rick's specific books for the cities or countries you're seeing. For an actual trip, you want the latest edition. If you're just going to London, Paris, and/or Barcelona, his books are sufficient; if you want to see more of their respective countries, look at other books besides his (again, raid your library). Rick's books are selective; he covers parts of each country in great detail, and completely omits other parts of the country. So, you'll want to learn about the places he doesn't cover, too.

Would you consider doing Seattle, San Francisco and LA in the same time frame?

Any suggestions?

Pick two