If the phone came bundled with a prepaid SIM card it very likely has a SIMlock and won't accept other SIM cards unless you get the phone unlocked. Except for that we have identical cell phone standards in whole Europe so there's a general compatibility between handsets and networks.
However there's no universal procedure on how to purchase and activate SIM cards - unless you are using a single SIM card and roam in these countries you will need to select a provider for each of these countries and find out where to buy them and how to activate them which can be a pain as you might be required to visit operators' shops to show your ID as part of the registration process. Also consider the costs of the SIM card itself and the minimum topup amount required. Depending on how long you'll stay that may not pay off.
Also be aware that in Europe you don't buy airtime volumes like in the US but here you simply top up amounts of money onto your prepaid account from which your usage will be deducted according to the valid price list.
As you seem to be new to this sphere, I think the most convenient way would be to purchase a
toggle mobile SIM card from the UK which allows you to request a local number from most of the countries you are travelling to through their website (so-called "toggle countries" which include Germany, Norway, Denmar, Sweden and Spain). I don't want to go too deep into the technical details but once you have assigned a local number from a certain country to your toggle SIM card calls in that country become very cheap and you will also have a local number under which people from this country can reach you at domestic rates, even if you leave that country.
Most notably the US are also a toggle country, so you can request an American local number under which people can call you during your trip at domestic rates. While you are in the toggle countries you won't be billed for these incoming calls to any of the local numbers you have obtained and once you are in the other countries not counted as "toggle countries" (Belgium, Portugal and Italy) with some modest £0.04/min.
Outgoing calls to the US are charged with £0.03/min from toggle countries, however from the other destinations the rate of £0.75/min is very high - you should consider let people call you instead of placing outgoing calls in order to keep your bill low.
For more information about toggle please see the corresponding thread:
http://www.prepaidgsm.net/forum/showthread.php?t=6765