How can I make money in the new career mode?

Do missions and milestones. The first few missions are really easy. The same for the milestones, you get most of your cash early game by completing them. Some of the strategies in in the Administration building could be helpful too.

The second thing you should do is make sure that every flight rewards you some science.

And as third you should attach parachutes to every decoupling stage. Sometimes multiple parachutes per stage are required. This way you're sure your decoupled parts are landing unharmed and gets you lots of your paid money back.

I find the best way is to double up wherever you can. I tend to take a group of the same contracts (this involves turning a lot of them down until you get near-duplicates).

Right now I have about 4 'place new satellites' lined up. I will launch 1 satellite to to accomplish all 4.

I also try to take contracts that I can address quickly (transmit science data from Mun Surface when I already have a probe there).

Generally any mission that requires me to visit a specific location I don't accept.

There are two answers I've found in my game so far: 1. acquire a penchant for low altitude plane flight and normal aviation and 2.(not yet tested, working on it soon) attach chutes to everything you jettison, and deploy it as you jettison them, so that your beloved rubble makes it to the ground safely.

As for 1, what I ended up doing was making a stupid move and upgrading my VAB and launch pad using all my early-earned easy cash, leaving me with the stock SPH and runway. Because of this, I thought my only option was to go into space and science the heck out of it like I did when the science update first came out, but this, as stated in the question above, turned out to cost an enormous amount of money and quickly had me reconsidering my progression path. In the end, I had built myself a mini science plane, capable of extremely long-distance flights and multiple safe takeoffs and landings. With the help of this, I designed it to make it easy to maneuver, while staying relatively stable and fuel efficient. Using this, I rejected any contract that took me higher than about 11k, which is more or less the ceiling of my craft's effectiveness, then took a level 1 pilot and level 1 engineer with me on "Visual survey of Kerbin" or "Gather science data from Kerbin" missions that fit the height requirements, and marathoned all of them in a single flight, repacking the chutes after every landing. Also take the easy missions when they come up, and build a purpose craft for it (AKA a pod with just that part attached to it, unless you need to take it up to a certain altitude, in which case I used rockets based entirely on solid fuel boosters which are pretty cheap) and complete the mission. Don't forget to do all the science you can wherever your missions take you, and recover rather than transmit it for full results (if you have to make a separate trip each time for the science JR and the Goo tank, why not do the same for all the other science modules?).

For 2, I'm still working on building my money and don't have enough liquid cash to risk on something like this, but I will edit this as soon as I'm done. Testing done, doesn't really work well as the pieces get unloaded and disappear when you get too far from them unless they've already hit the ground.

These are not by far the only answers there are (I think at least) so if you have a working strategy post an answer of your own!

A nice way to get some good money fast is to do tourist missions. Yes, you do lose money with decoupling all parts, but if you build a cheep orbiter with the Mk.1 crew cabin, it works nicely.