Both of your questions would depend on which phone you have and which network you are on.
If you have a phone with bluetooth then go for a laptop with a bluetooth adaptor.
Infrared connections are a bit hitty missy as you need line of site and a small distance between laptop and phone.
Your final method would be with a cable (either serial or USB).
Again, these depend on the phone you have (which must have a modem function/capability built in to the phone).
As for the mobile internet service, ask your current provider (Vodafone, Orange, O2, etc.) what they do for mobile internet connecting. They may be the best option.