What is RAM?
RAM, or random access memory, is a type of storage system known as volatile storage. That is, it only holds data while it has power. In other words, any changes you make to it, if you don't write it to a persistent storage device, will be lost at any time the RAM loses power.
What is a hard drive?
Hard drives, or disk drives (or hard disk drives), are a different type of storage system known as persistent storage. This means that the data stays even when deprived of power. CDs, DVDs, floppy disks, and USB thumb drives also fall under this category.
What happens when I boot, write to cache, or start new programs?
In order to maintain your system between reboots, the OS has to write what it has in the RAM into a persistent storage device, usually the primary hard drive. Conversely, in order to restore that information, or to start applications, it has to load that data into the RAM from the storage device. This even happens with live CDs, even though everything is running in memory, it has to first read everything from the CD (or USB drive). It also happens even with the tools/ideas cauon and Ugo have suggested, because at some point or another, data has to be read from or written to a disk in order to persist between boots (specifically Preload will not improve boot time).
If you have fast RAM, then your bottleneck will be somewhere in the read from/write to disk process, usually either in the speed of the data connection or the speed of the drive.
What can I do to improve the speed of these operations?
To improve the speed of all three of your mentioned operations, you'll need to do some hardware upgrading.
Get a faster connection. Parallel ATA connections (PATA, the gray ribbon in older machines) is going to be slower than SATA connections (Serial ATA, the newer, smaller cables). If you can and haven't already, upgrade to a SATA drive. If you go the USB drive route (outlined below), then make sure you're using a USB 2.0 (or, even better, speed-wise, 3.0) drive and port. USB devices and ports will clock down to the speed of the slower thing, so even if you have a 3.0 port, if the device it 2.0, you will only get 2.0 speeds.
Get a faster storage device. Platter-based hard drives are measured in RPMs, and generally come in three speeds - 5400 rpm, 7200 rpm, and 10k rpm. If you have a 5400 rpm drive, upgrade to at least a 7200. Additionally, Solid State drives (including USB thumb drives) are faster than platter-based drives. If you can afford it, upgrade to a solid state drive. Alternatively, you can install Ubuntu on a USB drive and run it from there (keep in mind, though, that such a set up will be limited to the speed of your USB port, so make sure you use at least a USB 2.0 port and drive).