Should I book hotels once or multiple times for weeks of staying?

As Calchas says, you might get the best price by negotiating with the hotel directly. If you tell them that you do not need room service -- or only need it once a week -- you may be able to get a significant discount as labor is one of the highest marginal costs for a hotel. The same goes for waiving breakfast, if they have it.

Also, you should try to determine if there's a convention in town during the time you are staying (you can ask at the tourist bureau). Hotel costs skyrocket during that time so if you can avoid it (by going out of town that weekend), you can also get savings.

Having been down this road before, I would carefully weigh the potential hassle vs. the cost savings. Having to change hotels (or even rooms in the same hotel) in the middle of a vacation (when you should be relaxing) or a business trip (when you should be focused on your objective) can be extremely distracting and inconvenient. While it may, in some instances, be cheaper to book multiple consecutive stays, you are also running the risk of complications which could leave you scrambling at the last minute. My advice, having travelled extensively for business, is to find someplace reputable and focus your energies on things more important than where you are going to sleep part of each day.

I used to have long hotel stays, and I can confirm you that it's possible to get a significant discount for long stays. Usually, I would book the first week online, and then have a word with the hotel manager a couple of days before departure. Hotels which are not in the top of a given city usually can't fill all rooms, so they will rather give you a discount than let you go to another hotel.

Another point is that hotels share their profits with the booking site when you book online, so they can offer you a better price directly.

In some cases you can get a discount from hotel award programs such as Club Carlson. With enough points you can get book a free of charge room or you can book at a discount with points (e.g. 50 euros per room per day for a room that normally costs 200 euro). Meals eaten at the hotel's restaurant charged to your room will earn you extra points.

If you plan to travel around without fixed dates to move from one location to the next, you can book online for the first location, refundable or non-refundable if you are really sure you want to stay there.

You can next make refundable bookings if traveling in popular times or wait with making a booking if it is almost sure to be possible to book last minute for the next town.

When traveling around, I usually stay in a city or village for two or three nights. In the past I would call the next youth hostel the night before moving on. These days I use a booking site and book my next place to stay, whether a hostel or a hotel, or an other kind of stay, when I am sure that I will be there or I get a refunable booking when I know I want to travel there but might not be sure about being able to make it.

As a tourist, you can often get good deals in business hotels in the weekends and mostly you will find there is a lot of space, but if you want to go tourist destinations in the weekends in spring, summer and fall, you better book ahead.

While hotels may offer a better deal if you stay more days, often three or seven, it might be cheaper for you to move on and find a new location. So do not just go for the cheaper room, look at the overal costs and whether you want to stay longer in that location.

That being said, take your time traveling. You will remember more when you travel less, take the odd day to relax and take a break from traveling by spending a few days in a (beach) resort, or go for a hike in the forest. Again, you can book ahead online but you can also walk in at a travel agency or a tourist information office and get a good deal or good advice from people in the area.

In short, there is no 'best' way of booking for a long travel, as a lot depend on the time of year, the kind of hotels you use, the locations and your personal needs.