There a a number of 'what-ifs' that come into play but essentially it CAN be that easy!
1. Assuming your PAYG Sim card is not 10 years old and is Data enabled (you can check this with your network -- if its not they can send you a new SIM for free)
2. Once you put your Sim in a new phone generally most providers download MMS and Web settings to your phone, or there is a quick way to get them e.g. Text ACTIVE to 2020 on O2.
3. If you don't have inclusive data on your plan surfing the web 'ad-hoc' can become expensive, some providers limit the charge, e.g. £1 a day whereas others charge by the amount of data downloaded.