I take it that by 'memory almost full', what you really mean is that your hard drive is nearly full.. The 'memory' in a machine (RAM) is volatile, when you power down, everything in it goes away. Hard drives are more lasting (but NOT truly permanent).
To upgrade a drive, my procedure is:
a> first run the Windows supplied utility programs (or aftermarket ones if you have and prefer them) to scan and defragment the existing drive. "Scandisk" is the Windows disk checking utility, and there is a defragmentation pgm also. They should be available thru: start>programs>Accessories>System Tools (this is for 98SE. The path may be slightly different for dif versions of windows.
b>'Back up' existing drive. This creates an exact copy of everything on the existing drive. Backups may be done with several available programs. Some come with new hard drives. Others are 'aftermarket' - i.e. sold by independent vendors after you buy your computer. My personal favorite is Norton "Ghost". Ghost lets you create an exact copy of your hard drive on writable CDs. You have to have a writing CD drive in your machine to do this - but you should anyway these days. If you are buying one, check to see that Ghost supports the drive you plan to buy. There is a list under Ghost on the Symantec.com web site.
It is possible to create more than one 'partition' on a hard drive. This causes the additional partitions to appear as additional drives. Your machine is set up this way if you have a "D:, E:, ... " drive in addition to the C: drive, and the additional drives are not CDs or DvDs. People often do this to keep data separated for easy finding of files. I have 6 different partitions on my drive: C: thru H:. Ghost will handle this by copying all partitions.
You should have a 'fairly fresh' backup at all times. If your hard drive 'crashes' - i.e. the heads grind into the magnetic surface on the platters - and this can happen at any time - everything will be lost. With a backup, you can fairly quickly restore your system on a new hard drive. Without it, you probably will NEVER get everything back - including your tax records, etc.
c> once you have 'backed up' your system - make an 'emergency start up floppy' using Ghost. Choose the option that provides CD drive support.
d> power down the machine, and replace the hard drive with the new one. Note the instructions with the new one as to how to set jumpers on the new drive, and whether the new cable (if you use it) requires setting the jumpers to 'cable select' (in this case the cable will usually be marked 'master' and 'slave' - or be two different color connectors at the drive ends).
e> Restart the machine using the Ghost boot disk you just made. Machine will come up in DOS running Ghost. Select 'image to local drive' function, tell the pgm to use the CD drive for input, put the first CD in the drive, and hit GO.
f> load each CD in turn to be read.. the screen display will show your progress. When you read and load the last disk, your system is restored.
g> Since the new disk is larger than the old one.. the restored disk will only be on part of the new one. (The image/restore process creates an EXACT copy of the old disk. The extra space on the new drive is not in any partition on the original disk - so it will not immediately be accessable. You can use "Partition Magic", or a bunch of other partition management programs to resize the partitions on the new drive to use the entire space of the new disk. You can make the partitions different sizes - depending upon how much additional storage you think you will want in each partition. If you change your mind later - it is easy to re-run partition magic to change the existing partitions and move the stored data.
When you get the new disk the way you want it - back it up again. This way, you can directly restore the drive with the new (larger) partitions in one step. I back my system up every 3 months - and store the CDs in a fireproof container. At 20 cents each.. CDs are a cheap,easy way to do this.. readily available too..