Yes RAID 0 will do this and increase performance. RAID 0 writes evenly across two drives accessing both at the same time. The upside to this is high performance by eliminating bottlenecking at the harddrive level ( good for programs that use heavy writing or alot of virtual memory such as games and movie editing programs). The bad side is if you lose 1 drive you lose 1/2 of everything literally and both drives have to be =. If they are not your hard drive will only be as big as the smallest drive x 2( so a 750 GB and 1TB drive would only have 1.5 TB of space total not 1.75 TB)
Since it writes the data evenly across two drives you will probably( extremely likely actually, but ive never attempted this on HDD with data on it for that reason) have to reformat and reinstall everything. You will also need a RAID controller. Some pcs already have these installed otherwise you can buy a RAID controller card ( usually pcie x1 slot). Your device manager should tell you if you have RAID controller installed already.
If you want a general idea of what RAID is about you can look it up on wikipedia. it will get you the basic idea of how raid works and the different configurations of them.
Edit to add: In response to an edit above me. There is nothing wrong with RAID 0. I run it on a gaming pc and have for some years. Newer drives and BIOS have S.M.A.R.T which will warn you in most cases of imminent ( physical) drive failure. Software corruption ( viruses malware etc) is 9 times out of 10 due to the users either not knowing how to protect their pc or being lazy.
Your risk of PHYSICAL failure is doubled because if either of two drives fails you will lose data( this is what SMART is for, so short of beating your drive with a sledgehammer, if you have SMART you should be alright). Data loss is not always an end all scenario. Their are recovery specialist who can recover data from drives in a majority of circumstance. Nor does it matter if you have two drives or one. That is illogical as drive failure or drive corruption would have the same overall effect on your system. If your data is extremely important to you and hard to or irreplacable it should be redundently backed up( two backups) no matter what drive configuration you run. I have an OS and games on my RAID PC. If it gets corrupted i lose some saved games and have to reinstall all the programs and update them again( something i would have to do anyway). It would literally be no different if i was running a single SATA.
And as far as i know you can run 2 different size( though not two types because their are SATA RAID and IDE RAID controllers) and/or brand drives. In RAID 0 it shouldnt recoginize the additional space on the bigger drive. Im not 100% sure on this aspect as i have always used two identical drives( and it is highly recommended you do as well) .