If you plan on passing around Paris we could organize a meetup around those dates; last Meetup was when Yvain was visiting, and there are plenty of nice things to see in Paris.
(That would make the Paris LessWrong chapter the "LessWrong tourist agency")
I was planning to pass through Paris and would very much enjoy that. I haven't settled all my timing yet but my current thoughts put me in Paris around the 22nd.
Read wikitravel articles on every country you are interested in. Also read travelblogs and/or forums. There are surprisingly many people that make make backpacking a lifestyle. Decent preparedness is very helpful. You should also brush up on where the EURO is valid and what and where the Schengen-space for easy traveling in the EU is. If you never been outside of the US read some other traveling reports as well. US passports actually contain a manual on how to use a passport, figure out why. Some different habits are important to know upfront., e.g. legal ages for drinking, strange laws that forbid stuff you are used too, and the other way round. Countries where Credit cards are not generally accepted, how to be polite in a way that the locals can deal with. Europe now also has cheap air-flight. Watch out for easyjet, ryanair, airBerlin, and many other lowcost carriers. Train travel also actually works for all of Europe, as do buses (Eurolines). My preferred mode is to pre-plan all the trips, with decent buffers for mishaps.
If you're travelling from the US, one trivial thing to bear in mind is that sales tax is included in the marked price in many European countries.
That is good to know, although traveling US to EU it will be a pleasant surprise. Probably more important to keep in mind the other direction.
About your request, more information needed: What are your vacation goals and constraints? E.g., do you want to visit touristy things (historical sites, museums, etc.)? Do you want to visit as many different countries as possible? Are you against any particular sort of sleeping arrangement, for example, hostels or couch surfing? Or is money not an issue compared to privacy and comfort?
About your specific situation, because I'm nosy: Which math conferences are you attending? Are you presenting?
I am attending a summer school in Sardinia and a conference in Benasque. I am not presenting.
I am interested in visiting multiple countries, and doing touristy things, but I'm not certain that those would be the MOST interesting things, so I'm interested in a brainstorm of things to do.
I had planned to stay at hostels, as money is a small issue and I'm not a big stickler for beds. I'd also prefer to meet people and do things together with them. Couch surfing would be better in every respect, except that I don't know anyone who lives in Europe and I'm not sure how comfortable I am sleeping in a stranger's house.
How long are you staying? If your journey begins and ends with those two events, and if you play on having any rest at all (i.e. you don't hate beaches and nature), a relatively direct road trip (recommended by train) along the Mediterranean coast Sardinia -> central Italy -> northern Italy -> southern France -> Catalonia could easily fill three weeks, and in July half of Europe moves there anyway, so you'll have no trouble meeting Brits and Germans. You could fit in one or two short visits by low-cost airflight to out-of-the-way destinations, but that's about it. Unless you seriously doubt you'll ever return to Europe again, I wouldn't try to squeeze in stuff like "two days in Paris" or "one day in Vienna", you'll just whet your appetite (and waste half as much time travelling as you would visiting).
Hostels are fine; get a Hostelling International membership and rely on it, if a hostel is affiliated with them it's almost guaranteed to not be too bad.
On a separate note, I hope this is not the only place where you're asking for advice. LW is a relatively small community, and for this type of advice it doesn't offer huge benefits over places like Reddit (where 'travel advice' threads are insanely common on near every European subreddit).
July 14th is the national holiday in France and July 21st is the national holiday in Belgium, expect shops to be closed and public transport might stick to the "sun- and holiday" schedule.
This site has a checklist of things you might need before you leave:
A lot of European countries (France, for example) have toll roads: before you hit a large motorway you get a ticket and when you leave you have to pay depending on how far you travelled. Other countries (Italy, I believe) require you to buy some kind of sticker, the more expensive the sticker; the longer you can drive around (ranging from weeks to months) Not only can this take a bite out of your money, expect traffic jams when everybody has to pay up, especially during the summer
Over this summer, I will be going to Europe to attend a pair of mathematics conferences. Because they are close together, I'm planning to spend the intervening time (most of the month of July) in Europe.
It is my first time going to Europe. I am very excited. I am certain that I will have fun.
I am equally certain that I will have more fun if I consciously attempt to maximize the amount of fun I will have.
However I know absolutely nothing about Europe; I have never been, I have few if any friends or family there.
So my question is, what should I do in order to have the best possible time in Europe?
I appreciate information that is useful for me specifically, as an American graduate student who has never been to Europe, who will be in Rome in early July and need to be in Barcelona by the end of July; the best ways to get around, benefits to being a student, events happening in July, visas that I might need. I intend to look for Less Wrong meetups occurring during my stay.
I would also appreciate general Europe information, such as excellent sight-seeing locations, the best way to buy food (I hear buying groceries in France is cheap and restaurants are very expensive), how difficult it is to move between different countries, the advantage to staying in one place for some time versus taking more of a tour of the continent.
There are certain types of information that I am not particularly interested in, but that I think would be appropriate to discuss in the same context. For example, what is the best way to find the cheapest flights or choose when to go on vacation (my flights are reimbursed and my timing is determined); what would make Europe an ideal vacation location, as opposed to Australia, Asia, South America, etc.
If people are interested in more details of my specific situation, I am happy to give them in the comments or private messages.