Stockholm has outstanding public transportation, clean, on time and safe. It may seem expensive compared to other places, but that is true for most things in Scandinavia.   Stockholm has  distance and express trains, subways, commuter trains, trams and even ferries.  The SL  web site has limited English, but if one uses google translate you can muddle through.   Resrobot.se has a better interface than the Sl.se site, and will give better information.  One can get to the most  places in Stockholm with out having to walk more than a 1 km.

Tickets can be bought at most subway and commuter stations from a human, but some stations only have machines, buses drivers do not sell tickets, One can pay for a ticket with a mobile phone, only  if it has been registered with SL.  A ticket for 2 regions is good for 1 hour and three regions for 1.5 hours.  If you are going to use public transportation more than three trips in a day, it is best to buy a pass card.  If you are going to be visiting museums and other attractions, the Stockholm one card is a good deal, plus it lets one use the city bikes. Transport tickets are available for  24 or 48 hours, or 1 month.  SL Tickets work on all Stockholm public transport, but not the city bikes,  tourist buses or the tourist ferries.  There substantial discounts for students, you have a student ID card.  If one is over 25  and from outside of the EU, they will probably not give a discount, but it is worth a try.  The fine for not having a ticket is 1200 SEK, and  ticket control does  check  tickets.   

There are buses at all of the airport terminals and boat harbors, For the Nynashamn harbor there is a commuter train nearby  that will take one into Stockholm,   If opting for a taxi, make sure to find out  the price before shutting the door.  

The subway and commuter trains stop operating from about 1:00 am to 5:15 am depending on the location. The are night buses will run all night long, but infrequently and not at all stops,  look for the sign that says nattbuss .  If you can't stumble home, keep an eye on the time. 

City bikes are the best way to see Stockholm, if you are only going around downtown as they can only be kept for 3 hours without a check in., They are a bit pricey for a 3 day pass compared to a season pass, but if the weather is good, this is the way to go. Djurgården has it's own city bikes that are a much better value, but once again they can only be used for 3 hours at a time. 

There are some beggers on the train, but they are not agressive,  There is a very robust cultural experience to be had if one is on the subway and trains after 23:00 on a Friday or Saturday. One can spot the rare phenomana of an outgoing Swede, but they are probably  drunk-  There are patrols on the trains and even late night its safer to travel than most other cities.