It sounds like a very fun trip! Do you also have people to stay with in the other cities, or only Rome? How "incredibly low" is your budget? In which places will you have a car or no car? How old are you? I ask many questions to maybe answer you better.
Of course everywhere it is easier to get around the major cities and cheaper with public transport, especially places like Amsterdam where parking is expensive or impossible.
Rome has a limited traffic zone that operates in certain hours I think, but more parking than Amsterdam at least.
It is helpful to know where you will or won't have a car or a ride. With a car, campgrounds outside of major cities are a great way to save money, but many of these campgrounds also have bus stops or trains nearby too. It is a nice perspective for someone who enjoys nature.
A city like Amsterdam is expensive at this time of year you are coming (how long do you stay?), and hostels get full, especially for the weekends very quickly. The whole city really gets full, even hotels, on the weekends so it is important to book things early sometimes. A hostel is probably your best option, or a place where you rent a room in a shared apartment. Air B&B has a few options that are very cheap, and I feel it is a little more safer than craigslist. At some times of year it can be very easy to find accommodations on craigslist too though. Here is a link to a list with some campsite info near Amsterdam, and as you can see, it is a very low priced option! But you must consider the cost of going back and forth to the city: http://www.iamsterdam.com/en/visiting/plan-your-trip/where-to-stay/campsites
This one is easy to get to I know: http://www.campingamsterdam.com/en/route
A hostel many people know for its lively social atmosphere is the Flying Pig and there are 2 within the city and also the beach hostel: http://www.beachhostel.nl/ The beach is the cheapest, but still easy to come and see the city, very good for meeting many traveling people and partying, and a more natural setting. Also it is usually less booked than the others. I hear a mix of opinions about the downtown flying Pig. Some say it is not clean, some say it is, but I know many who say it is where the party is for young travelers. Myself, I have been in the Bulldog hostel (with friends, or at parties so I am familiar with the hostel rooms) and am impressed by how clean they keep it.
Our city parks are beautiful too, like the Vondelpark.
For small items it is easy to store at the Centraal Station, but there are also some independent businesses too. I do not know how they work with weekly rates or if they have a limit on time you can leave things. I might be wrong but I think the airport has a time limit to leave luggage. The problem is the pricing! Maybe the place where you stay can watch your things for you if they are not so big until you come back?
I have heard positive things about here from guests to the city, but I have not used them myself: http://dropandgo.nl/ If I am correct they have different prices for longer periods, so I think they will work with you.
Another place looks a lot cheaper, but again, I have not used them or know anyone who has, but here is the link: http://www.allsafe.nl/english
It is a friendly, lively, open city, with many foreigners and travelers speaking all languages so you will have no trouble meeting people to have fun with. The atmosphere in the hostels and the campgrounds is enough of a social one that you can be surprised at the connections you make. The coffeeshops too, if that is a thing that interests you. Sometimes if I sit in one I will leave with many "new friends." Some of these have become real friends and even come back to visit me or host me in their home city. In my neighborhood it can be common to see people having street parties on weekends, or make friends in the bars (I stay in the Jordaan area).
-to be continued-