You are correct, your OS will only boot up faster with a solid-state drive. That being said, if you perform any operating system related task that needs to retrieve data from the drive, it will be much faster then if your OS was on the HDD. That being said, if you only care about loading subsequent programs (or logistically prefer keeping your OS separate), you can easily keep the SSD as a secondary drive, only using it for certain programs/tasks.
The whole point of a solid-state drive is to decrease application loading times, and that's it for most users. This is more due to the lower seek time rather then the faster transfer rate, which makes it more like RAM (good sustained and random transfer rates). In fact, some users would be better off getting more RAM then a solid-state drive - but that always depends on your needs as the user.
That being said, the main storage device (SSD or HDD) is always the bottleneck of any computer system. While SSDs help to alleviate this bottleneck, new ones are still only ~1/40th the speed of RAM. For example, some memory bandwidth in newer computers has reached over 20,000 MB/s versus some new SSDs which top out at just over 500 MB/s.
You can also use it for the increased sustained transfer speed, but that only applies if you deal with very large file transfers with, for example, video encoding.
For the fastest experience with your computer, install your OS on the solid-state drive, but do remember to make frequent backups. Yes, it will mostly affect just your load times, but again, that's why you put data on a solid-state drive in the first place. They are not meant for storage of large amounts of data, just your OS/programs/games only.
Finally, since most SSDs have a smaller capacity then hard drives, you may wish to "lighten" your OS install by not installing as many packages (if you use a package-based Linux distro), or using a utility to remove components from the installation media (if you use Windows).
If you're using multiple resources on the same hard drive, then this is normal behaviour. Hard drives are big, mechanical devices which are good at one thing - only doing one thing at a time. You can only read/write to a single sector at a time, so attempts to use the hard drive simultaneously usually result in thrashing. This isn't a side effect of anything other than the hardware working as designed.
If you're doing multiple things on a mechanical hard drive at once, you might find better results by performing these actions one at a time. If you're talking about file transfers, considering replacing the Explorer file copy handler with another program which supports transfer queuing (such as TeraCopy, which will also allow you to pause the transfer if you need fast disk access temporarily).
You can also help to mitigate these effects with the use of a solid-state drive, but it is no guarantee - it only helps because of the drastically reduced random access time. You can still have thrashing with an SSD, and an SSD still has the same limitation - it can only read or write to a single sector at a time.
If an SSD isn't the route you want to go, consider using a RAM disk, performing any data processing in-place in memory, add additional drives (not partitions) for each task, or performing I/O intensive tasks on non-OS drives.
Best Answer
You ask why SSDs don’t necessarily improve the speed of running applications compared to a traditional HDD. The reason is disk access. SSDs do read and write much faster than traditional hard disks. However, unless the application is heavy on disk I/O (reading and writing to the disk), the benefits of the SSD are lost. Even in cases where there is some writing to disk, the OS has a disk cache. This means writes are written to RAM and eventually flushed to the disk at a later point. The disk cache significantly improves the speed when using traditional HDDs.
As for swapping, yes, SSDs will perform faster, but you probably won’t notice the difference. Unless you are severely starved for free memory and the OS is heavily swapping, you will not notice the speed difference.