If you have a drop ceiling like the one in the picture you may want to look above it to see if you have access to plumbing and electrical up there. If you do, you may want to keep the drop ceiling or at least figure out some way to keep access to the wiring/pipes.
Remodeling the basement the right way probably means tearing out the particle board. I would want to do it just to make sure that I didn't have any problems with the framing behind it and there will never be a better time to insulate, re-plumb or re-wire. Tacking sheetrock directly to the particle board will probably not create any problems, but installing casing and trim around doorframes will become much more challenging.
The second pic you posted is probably just sheetrock that has been painted, maybe a bit of texture over it. Nothing terribly special goes into that look.
Installing sheetrock is really pretty simple. Finishing sheetrock is tedious and definitely has a learning curve, but it isn't surgery. There isn't any reason that you can not do all of this work yourself. Whether or not you can achieve at least a satisfactory result is a function of (i) if you have any native handymanliness, (ii) how precise a worker you are and (iii) how confident you are and how quickly you can learn. To get a better idea of whether or not you are up to the task, check out this site:
How to Install Drywall | Hanging Drywall | Sheetrock Installation | Installing Sheetrock