Depending on where you are there may or may not be a public bus system, if you are lucky there will be a big public bus system. This equates to free rides.
Tell the driver upfront you dont have enough to pay the full fare and they'll usually be fine with it. Be curtious and expect to bring out the misery in few, in this case just wait for the next bus with your thumb out. Unless its an automated system where the driver has to put a few coins of his own in to let you on.
Check the route timetable and tell the driver the name of the station 2 stops down the line, pretend to fall asleep if you want to make sure you're not bothered, otherwise just stick headphones in, read a book or be in an indepth conversation, if the driver stops the bus and confronts you (which will almost never happen) just say oh you forgot where you were and get off or if you want to carry on going tell the driver you forgot to say you changed your mind and pay to get where you want to go. Dress like a clone and hiding your face helps the driver forget you.
You'd be hardpressed to meet a bus driver who'll leave you stranded on the last bus home (especially if your a girl) if you give them a big sob story, but they will have herd it all before so keep it short.
Flash child tickets and move on.
Transfers are often color coded for the day of the week, meaning a day old transfer is pretty much out, but most drivers won't check the times on a transfer if it the right color. This is not true in most modern cities where you have to swipe a magnetic pass or deposit money into the machine. Check garbage cans near transit points. Another method is to ask people getting off the bus if they will give you their day pass, or even offer to purchase it for 1/2 or less cost. You can also sell your ticket in the same fashion to get a few extra dollars for the day. If you are feeling more legit and younger looking many bus systems have a reduced fare for youth.
If you are lucky enough to be in a city that allows people to enter at the back of the bus just walk aboard and take a seat. This works best on the busier routes and there are others getting on at the same time. The traffic pigs usually do a sweep at the beginning of each new college semester to 'persuade' new students that it isn't an option. Best buy a ticket then and at the beginning of the month, especially if the first is on the weekend. Corpgov will even hassle the three-piece suiters about not having their monthly pass updated. Always keep enough money for the fare on hand so you can talk your way out of it. If you're dressed in corporate camouflage you can use the "I haven't had enough coffee today" line with a moderate degree of success.