Yes. What you need is a utility to Clone your old drive to the new one, with a couple adjustments. Depending on whose new HDD you buy, you might get that cloning software free by download.
IF you buy a WD unit, download from their website Acronis True Image WD Edition. It will make a clone TO a WD drive and not to anyone else's. If you buy a Seagate HDD, download from their website Seagate Disk Wizard. It appears that both of these packages are actually customized versions of Acronis True Image. It is very good software that does LOTS of things besides cloning, so be sure to get the manual file with it and READ it! Then you'll understand how to use the cloning tool, and how to find its options. If you buy another company's HDD, check their website for a similar cloning package. All these cloning tools make moving from one drive to a new larger one real easy, which is why the makers of new HDD's offer them free - to help you buy their units.
To use one of these you install it on your existing HDD. Then you install your new HDD in your machine, but don't expect it to show up in My Computer. It needs some preparation before Windows can use it.
You run the software utility and go to the menus for making a clone copy. Make VERY sure you set the SOURCE unit to your old drive, and the DESTINATION unit to the new one - any stuff on the Destination drive gets wiped out, so it better be the new empty one! What the clone process does is copy EVERTHING from old drive to new, putting important files in the right places, and ensuring the new drive is bootable. The idea is the new unit should completely take over as your boot drive called C:, and you can remove the old drive entirely if you wish.
To prepare a HDD for use by any OS, two steps are needed. First you must Create a Primary Partition - this is just a chunk of the HDD's set up to be used as one "disk" with its own letter name. "Creating" it means writing a small special file called the Partition Table to the beginning of the HDD; this file contains the specs for the location and size of the Partitions on this HDD. You can have more than one Partition if you choose, OR you can tell it to make just one Partition of any size, up to the entire HDD space. The second step is to Format this "drive" you've just created. This establishes the File System being used on the Partition ("drive") and writes to it some key system files. Once these are done, the OS can use the Partition as a drive with a letter name.
Now, YOU do NOT have to do any of this! Both of these jobs will be done for you by the cloning software. But you need to understand a little bit to understand how some options are set.
You'll need to watch the screen prompts for where to intervene and choose menu options - that is why I said to read the manual. Most such software I've used has a default choice I don't like, and I suspect you will want to change it. When it goes to establish the first Partition to hold the clone copy of your old HDD, by default it wants to make the clone the same size as the old small HDD, leaving a whole bunch of Unallocated Space you can use later to Create more Partitions to be used as separate "drives". But most users actually want ALL of the new large HDD to be ONE "drive" with lots of empty space for the future. So you need to tell it to change the Partition size it plans to create to what you want - maybe all of the HDD. The other option defaults usually are just fine, like:
1. This should be a Primary Partition, not Extended.
2. It should be bootable - you will be booting from it!
3. The File System to be installed should be NTFS unless you know you need the old FAT32 system for some reason.
4. You can tell it to do a Quick Format. This will do the preparation job in about 15 minutes or less, the go on to doing the copying process. The disk will be perfectly usable.
5. Alternatively, you can tell it to do a Full Format. This does the Quick Format, and then exhaustively tests every sector of the HDD, marking off any faulty ones so Windows never tries to use them. This process can take several hours, and THEN it can do the copying. Many people think this is not needed on a brand new drive, but some do it anyway just to be cautious. Your choice - all it costs is patience.
After the clone process is finished, I recommend you shut down and disconnect the old HDD, then plug the new HDD's data cable into the mobo port that the old one was on. (That assumes they are the same drive type, like both SATA. If not, forget this idea.) Leave the old HDD disconnected, whether it is left inside your case or not. It is a perfect backup of your system right up to when the new drive was installed. If anything goes wrong, you can go back to it. When you boot up, the machine should boot just like it always did, except that you will find your C: drive has a LOT more Free Space.
After you're satisfied that everything is working properly, you can decide what to do with the old HDD. Maybe wipe it clean by re-Partitioning and Formatting, so it becomes a data storage drive. Or maybe move it to an enclosure and make it a portable external HDD. Either way, that cloning utility probably will have tools to help with that.