As long as you don't need to physically move the larger drive to the new computer this should be relatively simple. You say you have saved the files to a disc, is that a CD/DVD RW disk or an external HDD? If it is opticaland the new machine has a suitable optical drive you are good to go, I know many laptops have now dropped optical drives. The same applies really if it is an external HDD, since you will just need to plug it in to a USB port.
Once you have the new computer you just need to copy the saved files over the the internal HDD of the new machine, placing the photo image and other document data files in matching locations on the new system. Then you just copy across the installation programs for the software that you want installed on the new computer.
When it comes to installing the programs you probably don't want to be bothering downloading the programs again for older software, just go with the files you have. If any of the software you are using is current though it would probably be best if you went to the software publishers site and ran the install over the net again. That way you will be starting out with the latest version of the software. If you use an older install file all that will happen is that you will do the install, and then immediately have to do the update, which is a bit of a waste of time IMO you might as well just do it from the start.
Just remember that you will be installing the software from scratch on the new system, so you will need to have all of your product keys available to enter when requested.
If the larger HDD is in the older computer and you really need to change them over then I really think you would be much better off taking ti to a local computer shop, and paying them to do the job for you. I say this because if you get it wrong there is a very good chance that you would end up having to pay someone to fix it anyway. That is very likely to cost significantly more to fix, simply because fixing ti will take them longer than just doing the swap for you will. You also won't need to buy the hardware that you would need to do the swap, so that will also help offset part of the cost.
once you have got the new system running I would look into getting an external 2.5" SATA enclosure with a USB3 interface. You should be able to find one that will allow the drive to be powered from the port, but I would still look for one that also allowed and external power supply. It is quite easy to plug the old HDD into the enclosure so that you can use it as an external drive. When you first plug it into the computer though you will need to format the drive to remove all the old data from the drive. Formatting is really the best way to do this with an old system drive. I suggest using an enclosure with USB3 as a minimum, even if your system only has USB2, since the drive will work OK in a USB2 port, but won't slow things down later if you need to plug it into a computer withUSB3.
I would agree with Dave's post above, if as it seems the "new" computer is actually a bit of a downgrade I wouldn't do it. If all it is effectively going to do is get you a working battery then you would be much better off changing the battery for a new one in the existing computer. This actually has advantages, since a secondhand computer will also have a secondhand battery, with undetermined remaining life. A brand new replacement battery on the other hand is sure to have a decent lifespan ahead of it, and will probably outlive the rest of the machine at this point.