So far the answers could be confusing. Maybe the Op is asking about some kind of extra cost for accessing the internet.
Op, if you are somewhere that has free wifi (McDonalds, Barnes & Noble, Starbucks, various hotels, schools, restaurants, etc), you can access "free Internet" via "free wifi." In other words, you could just buy a Mini and have no other costs for accessing the Internet as long as you can tap into free wifi provided by someone else.
If you have a home Internet connection and maybe wifi set up at home, you can access Internet from your own "free wifi" at home. No additional costs there.
If you are wanting to access Internet somewhere and can't locate any free wifi, then the LTE version can access Internet via a cellular connection. Those require you to buy a tier of data from a cell provider like AT&T or Verizon, etc. You can do that on demand (meaning you can turn on cellular-based Internet whenever you need it and only if you need it).
If you are somewhere that sells wifi (such as most of the airlines in flight, some airports, etc), you can also buy some Internet access from this "nonfree wifi".
In all cases, there is no additional fees from Apple for Internet access. It's just a matter of whether you can source free wifi or not as offered by other companies... or if you can't and then need paid cellular data tiers or not.