Yes, one of the reasons for using 2-layer PCBs is to avoid the jump wires. On a complex board there would be lots of them in all directions. Your layout doesn't need the second layer, though; it's still simple enough to be routed on a single layer without wire bridges. If you get stuck you may have to move some components to another position, but it can be done.
I would suggest to try it instead of going the easy way of the 2-layer board. It's about practice. If you don't succeed in doing this on a single layer, you'll never be able to do a more complex board on a 2-layer PCB either.
A professionally made multilayer PCB will have copper-plated vias to connect the top trace with the bottom trace. If you plan to make the PCB yourself you won't have the through-metallization, but since you only use PTH parts you can solve that by soldering the wires on both sides of the board. For some components the wire may not be reachable on the component side. Just make sure there's no trace to the hole on that side then. If you would need vias apart from the components' mounting holes you can solder a thin wire on both sides of the hole.
2-layer PCBs will often use both layers for routing, and even component placement. On densely populated boards this may not leave much room for copper planes, like for ground or Vcc. If these are really required designers often will go to a 4-layer board. A ground plane not only helps in the design of HF designs, but ground is often the net with the most connections, so it takes a lot of routing away from the outer layers as well.
Today most PCBs use SMT technology, and SMDs will often be placed on both sides of the board. First the components on one side are fixed with glue dots, then the board is flipped and the parts on the other side are placed. The glue is needed to avoid the components falling of when the board is turned upside down. Some glue dot machines can place up to 50000 glue dots per hour, that's 14 per second!
The wave only contacts the bottom side of the PCB.
Once upon a time, a wave solder machine was used to solder SMT parts on the bottom side of the PCB but this is not used as much anymore in favor of more modern techniques.
Here is the rough process for soldering PCB's with SMT parts on both sides, and through hole (TH) parts on the top side only.
A. The bare PCB is turned "bottom side up". A solder paste is pressed through a stencil and onto the pads of the PCB. A pick and place machine places the parts onto the bottom side. The PCB is ran through an oven (hot-air convection or an IR oven) to melt the solder and attach the parts.
An optional step is to place a small drop of glue under the parts. Solder paste first, then glue, then the parts are put onto the PCB and soldered. This glue helps keep the parts from falling off during a later step.
B. The board is turned over (top side up), and the same process is repeated for all SMT parts on the top side of the PCB. By that I mean solder paste, parts placed, then through the oven. No glue is needed.
During step B, the parts on the bottom of the PCB don't fall off. Obviously if they are glued down then they are stuck there, but most companies do not use glue. Without glue, the surface tension from the molten solder is enough to hold the parts in place. Some parts, particularly heavy parts without many pins, might not work with this technique since there is not enough surface tension to hold the parts on.
C. All of the through-hole parts are then placed on the top side of the PCB. A solder pallet is attached to the bottom of the PCB. The PCB is ran through a wave solder machine to solder all the TH parts.
Note: A solder pallet is basically a shield to protect the SMT parts from being removed in the wave. They are custom made for each PCB, and have holes and contours to expose the TH parts while shielding the SMT parts. The PCB must be designed with the solder pallet in mind, since you can't place the bottom side SMT parts too close to the TH parts and the SMT parts cannot be too tall.
A relatively new technique for TH parts is to skip the wave solder machine completely. Back in step B, solder paste is placed on the TH pads (and in the holes) and the TH parts are inserted and soldered in the oven with the rest of the SMT parts. Some companies, like Motorola, have gotten rid of their wave solder machines in favor of this method. But most companies still use the older technique of using a wave solder machine with solder pallet.
There are, of course, many variations of this entire process. I have just given a simple and brief overview. But it is fairly consistent with the way current manufacturing processes work today (things were different even just 10 years ago).
Best Answer
While I have to agree with the previous answers - you are indeed a lot better off in the long run if you route manually, I feel your question has not really been properly answered.
A quick-and-dirty workaround for you might be
Now, before the rotten-egg-throwing sets in, a few extra words why you should not do that.
While Eagle takes quite a bit of effort to get the hang of, it is definitely worth practicing these things on simple designs. As you advance, you will get to the point where you have to route manually, because some signals must be laid out in certain ways. There might still be workarounds for each specific problem, but you will never have practiced to place the components in a way that it is possible to route with minimal effort and losses.