Good choice. While in Italy or England you will be roaming. That is not the dirty word it once was. Both countries have extensive and excellent mobile service. When you arrive in either country, you will receive a text message from T mobile telling you welcome to the United Kingdom and info that calls cosr $0.20 jper minute etc. The name of the local carrier you are roaming on will be visible. In Italy, for example it might be TIM. In the United Kingdom, it might be EE UK or whatever. For the most part in those two countries, it usually does't matter. Not necessary at this point to go through a whole song and dance of how to manually choose a roaming carrier.
One caveat. In general up till now, roaming data has been somewhat slow. T Mobile had a special this summer for high speed data in those two countries among others. No indication that it will be extended but we'll have to see. But while you are in your hotel, there is a good chance you will get free wifi and there are many hot spots I know of in the UK (Mickey D, Starbucs, Pret a manger) so you can always pop in if the data is too slow.
Your friend will pay for texting to your us number (he or she dials +1 followed by your area code and number) to send a text. He or she will pay whatever his or her carrier charges for international texts which is totally dependent on his or her plan. You pay nothng to receive or send texts. To send a text while you are in Britain to your friend assuming he or she is in Britain you can use +44 prefix followed by his or her local number omitting the lead 0 if there is one or by dialing the number with the lead zero and omitting the +44 but it's easier to program all your numbers with the + and the country code omitting the zero.
Hope that helps.