The best way to explore the subsurface oceans of Europa is to see what comes out from the vents. Basically, along each crack will be a small amount of ocean water. Take a sample, and you could find out a lot about what is below the ocean. One simply has to land on a recent crack, and you should be able to find something out.
Actually getting in to the ice is far more problematic. There are two primary challenges associated with this. The first is simply to get below the ocean layer to begin with. The second is how to communicate with the probe after you have gone deep. Both are problematic.
Let's start with the first one, getting underneath the ocean. The easiest way would be to find a crack forming, and work your way in to it. This is actually quite difficult to do in practice, mostly because you have to know where the crack is going to form before you land, and these cracks change location every 3 days or so. And they are pretty small, for the most part, you might have difficulty fitting a probe in to one. Plus they make the communication more dangerous, as you'll see later on.
The second way is to take a portable heat source and melt the ice away. Basically, this would be a novel use for an RTG of sorts, radioactive heat could melt a hole. This could take a long time, but would work eventually.
Lastly, a hole could be drilled. The most challenging factor with this is the fact that even the best drill bits need to be changed on a regular basis for mining operations. Let's say you could get 300 hours out of a drill bit, at a rate of 3 m/hour, leaves only about a km before the drill bit needs replacing, and that's probably optimistic. Who is there to replace the bit?
Okay, assuming you can get through the surface, what do you do there? Basically, this is in the same realm as submarines, except through ice as well as water. There is a whole Wikipedia article on submarine communications. Method could include buoys to the surface, running a long cable, sound waves, or very low frequency waves. The cable or accustics are the best ideas, ideally there should be a combination of both of them, to give some redundancy in case something fails. Keeping the cable intact, would be difficult due to the moving ice, and acoustics would probably be off a bit as well, but they could be done.
Bottom line is, the only cheap way to study the oceans of Europa is to see what is on the surface, along with technologies like RADAR to see what lies beneath. Truly exploring it would be amazing, but incredible difficult and expensive. Someday we'll get there, however.