Sorry I've taken so long to get back to you about where to go in South America, I'm a lazy sob. I’m going to do go country by country so that its not completely unreadable.
Transport wise, if you want to do this on the cheap, buses are the way to go. Colombia is the most expensive country for buses, but still a lot cheaper than flights. Most of the buses are fairly comfortable, and make regular enough stops. But expect bus trips of 10-20 hours. Its generally not that bad though, and the varying scenery that you see as you drive makes up for any boredom, and makes this a far more attractive means of transport than flying.
Ok, so you mentioned that you wanted to start South America in Columbia, Cartagena. This is an awesome city, and I would highly recommend staying in El Viajero hostel. They organise some sort of party almost every night of the week, and there's just generally a great atmosphere and its well run. Link:
There’s a somewhat decent beach here. Its mostly just an awesome city to walk around. It is divided into the old and new town, seperated by a circular wall which was built back in the day as a fort, and pretty much encapsulated the city.
Also up in the north of the country, there is a fantastic national park called Tayrona. You can stay overnight here cheaply, as long as you dont mind sleeping in a hammock. Alternative options are staying and keeping your bags in a nearby town, either Santa Marta or Taganga. These are both beach towns as well. I’ve heard stories of a hostel in Taganga where the owner comes around and hands out coke to people staying there. I can’t confirm this as I haven’t been to Taganga. Tayrona National Park really is worth checking out though. Its kind of prehistoric looking in terms of the scenery and beaches, really beautiful.
From Tayrona, go to Medellin. This is the city which was so dangerous a couple decades ago when Pablo Escobar was causing havoc. You can actually do a tour of Pablo's life. There are 2 tours, one of which is run by Pablo's brother. I've heard the one not run by his brother is better. The party life is decent here, and its famous for the women getting silicon in both their tits and ass. Also, almost everyone was doing coke when I was here. Not the best/most interesting city apart from that though.
Next stop, Salento. This is a chilled little city set in the mountains, beautiful views and also one of the spots where Colombian coffee is made. 2 Things to do, coffee plantation tour which takes a few hours, and a hike up into the valley, which pretty much takes the whole day but its stunning. Recommended.
Last stop will be Bogota, I didn't do too much here because I didn't have much time. It’s meant to be decent to party, and there’s a gold museum which I heard was worth checking out, as well as the monseratte.
I’ll try get a write up going for the next country asap. Hope this helps OP, I trust you having an amazing time! Give us an update sometime.