Well, it all depends on where you're traveling to, what you want to see, what kind of traveler you are and what you want to get out of this trip. When I lived in Europe I made all the travel arrangements myself and I didn't have any problems, but you need to research the transportation systems and situation in your countries before you travel over there to make things easier for the both of you to keep your sainity! Personally I'd make the traveling arrangements myself, because you'll get a better feel for Europe and what it's like, while with a traveling tour, you're going to be disconnected from everything and everything is going to be planned out for you guys by someone else, while if you do it yourselves, you will have more personal control, if that's what you want to do. Now to get around from country to country, you could go with rail and use a Eurail pass, but I know that all the other backpackers on this website are going to protest when I say it, but I don't recommend the Eurail pass, but of course with my situation (I lived in France for nine months) having the Eurail pass seemed silly, but the thing with the Eurail pass is that you pay this big price and then on top of it you have to pay a discounted fee for everywhere you go. On top of that, you might have a lot of train connections to do from country to country, and for first time travelers to Europe the train system can be very confusing and it varies from country to country. Also, but of course, this is my personal opinion, the train systems in most of Europe are good and I did use them heavily especially when I was in France, but I think they're a bit over rated so to say. When I traveled around, I used Eurolines, which is a bus system known all over Europe and they are very reliable and safe and are reasonable. (go to http://www.eurolines.com) They have a flexible pass for like 140 euros that you can use for a month and you just pay that fee and you can travel whereever you want to for that month without paying another fee! That's what I did and it was worth it! You could also book flights with Ryan air which is a discount airplane company which flies to a lot of destinations in Europe for a very low price. So what you could do is book your big flight to Europe to your first destination and I think you might be able to book your eurolines pass ahead of time and you could find out where you needed to go to pick it up (keep in mind, they go to certain cities so make sure that your first destination will be where there is a Eurolines office, center etc in order to pick it up) and just pick up new tickets as you go or you could book your big flight over to Europe and then book ahead of time flights with Ryan air to the cities and destinations you wanted to go to. If you think this is too much for you and your boyfriend to handle and if this isn't your style, than book a Europe package with a travel agent.
In regards to money, it depends on where you want to go and what you want to do. Remember that not all countries in Europe use the euro. The UK I believe still uses pound sterling, but in Portugal, Spain, Italy, France, Germany, Belgium, Switzerland (?), Luxumbourg, Denmark, the Netherlands, Finland, and I think Sweden too all use the Euro. All the Eastern European countries have their own currency but will eventually switch to the Euro or not and Norway is not even a part of the EU so they don't use the euro but their own currancy. So it depends on what currency in what what country you're using and what the exchange rate is as compared to your home country and the country or countries you're going to visit.
In regards to money and budgeting it well, it depends on where you want to go but here are somethings that I did that seemed to help: if you can stay in a hostel, stay in one because they are cheaper than a hotel but if you have to stay in one, stay in the cheapest you can. If you and your boyfriend are students and under the age of 25 buy an ISIC card, because this will help you to get students discounts in museums, etc. I know that mine was very useful when I lived over there. Also to cut down on costs, you could prioritize on the museums and all that you really want to visit that have a fee and see those instead of trying to see all of them. Also a lot of European cities offer city cards, which will not only allow you to get discounts, but also free entrance into museums, attractions, etc and also free run of all the transportation in town. Eat street food, buy cold cuts and cheese from local delies, buy nice breads and savory tarts and pastries from local bakeries, get preprapaired foods and or ingrediants from a local supermarket or market to cook up at your hostel or heat up at your hotel, and go to restaurants away from the main tourist areas, and you will have better, more authentic food anyways. Also do as much research of your destinations and countries ahead of time so that you know what to expect in regards to prices and just everything in general (transportation, etc) so that you don't make heartbreaking mistakes that force you to waste money on accident (take it from experience).
Well, Melissa sorry for the long reply for I hope that this helps. whatever you and your boyfriend decide to do, enjoy your trip to Europe and whether you go with a tour or if you do this yourselves you're going to have a blast I hope; I know I did when I lived over there. Good luck and enjoy!