As you indicated your first port of entry is France, you should apply for a Schengen visa with the French consulate. Incidentally, they are the best at granting longer-term multi-entry visa.
It's been a while since I needed to do this, but at the time, there was no difference in price between a single- and multi-entry visa. What you should do is state in the application form your intended itinerary as entering France, staying in France, then on to Sweden, then back to UK - and then later travelling to France again. You can also submit a cover letter explaining in more details why you are requesting a multi-entry visa.
It is quite common for French consulate in the UK to issue multi-entry visa valid for 6 months to first-time applicants. Many other countries would automatically issue a single-entry visa for you.
Once you complete your first trip, as long as the visa is still valid (date-wise), you are allowed to visit any other Schengen country, provided you return to the UK by the expiry date on the visa.