Hello Denise, welcome to the forums!
I'm willing to bet that 95% of the people that you consult here will recommend that you travel independantly of the touring companies - bike tourists are independant beasts by nature, and nothing thrills them more than wandering and discovering a new place without the hassle of someone deciding their itinerary.
You sound like you've just started thinking about this, so maybe I can help you get some perspective. For starters, you want to decide whether it's worth it for you to go with a touring company, or do this independantly. There are pros and cons to each.
The touring company pros are:
You don't worry about the logistics. Like you said, all bookings, shipping of your gear, taking care of where/when to eat, where to sleep etc. You don't have to know how to fix a flat tire. They often provide knowledgeable company during your trip. Another big plus is that they know some of the prime cycling paths and countryside, and so they can put together an itinerary that lets you see the most beautiful landscape and as many major features as possible with a limited amount of time in a new country. Many people choose the touring companies when they want to maximize the distance/sightseeing they want to do in a short time frame, or if they've had absolutely no cycling experience and are nervous about jumping in without help.
Touring company cons are:
Like you said, cost. You can tour much, much more cheaply when you plan your own tour. You can stay at cheaper hotels/hostels/campgrounds/free camp where appropriate. You can eat out of grocery stores, and save money up for a couple of really nice dinners without breaking the bank. I've already mentioned that if you tour on your own, you are much more flexible about your itinerary. Heard a local mention a wonderful place about a day's trip away from your destination? Well, just turn your bike around and go there, hassle free. Speaking of locals, they tend to be more open and easier to interact with when you're touring independantly. People admire independant bicycle tourists, and I've been absolutely bowled over by the warmth that strangers have shown me on a tour. Tour groups, by their very nature, isolate you from the people that live and work in a country.
So there's a general breakdown. Like I said, I (and most anyone here on these boards) is probably biased toward independant touring. Your decision about which route to take is a personal one, so take a long look at what you want to get out of your tour so you don't end up frustrated at the end of it. One quick note here: inexperience doesn't mean you can't tour on your own. My first major tour, like yours, in Europe, and it was my first tour ever. It was a blast.
Okay, if you decide to do the tour on your own, you can think about going with a friend/finding a touring buddy on these boards. It can take the edge off the nerves when someone's travelling with you. Your logistics will include: eating, sleeping, and making distance so you can see the country.
Eating: Usually the best places to get cheap food are regular grocery markets, however in smaller Italian towns (the non-touristed ones) you can often find good meals for a pretty low price. Note that in Italy and Spain there is a siesta during the middle of the day when all sotres shut down.
Sleeping: Booking places to sleep (the cheapest non-camping route you can go is to book hostels) isn't a huge problem, as long as you call them a couple of days in advance. After you've booked your first one, you'll be an old pro.
Making the distance: this involves your bike, and your travel itinerary. You can find a lot of information on what kind of bike to bring on these forums, I'll let you do the search for that. As for whether to bring your own bike or rent one, that also spirals down to cost and logistics. It's cheaper to bring your own bike (generally speaking), and you don't have to return it to a certain place at the end of the trip. However, if you bring your own bike you have to pack it in a bike box to get it on the plane (your local bike mechanic will help you with that), and put it together when you land. As for your travel itinerary, your travel route depends on the types of daily biking distances you plan to ride. As completely inexperienced, UNTRAINED riders, My travel partner and I rode between 30 and 60 km/day for the first week of our tour. We took a couple of rest days, and we also spent a lot of time stopping and starting, taking photos, having lunch, seeing the sights etc. At the other end of the scale, it's not unusual for a trained, dedicated touring cyclist to ride over 100 km/day. Distances depend not only on the rider, but the weather (hot/humid/headwind = lower daily distance), the road conditions, and the terrain (you won't make as much distance up and down mountains as you will on flats), so take that into account when you plan your itinerary.
Okay, so I hope this gets you started with your decision-making. Feel free to post more questions on the board as new threads and I'm sure you'll get some replies. Good luck with your first tour!