If you went through a lot of film, you can expect to take a lot of digital pictures. That means that you will eventually have a data problem. Too many pictures to be able to find the one you want, unless you develop a digital management system.
I think the Adobe Lightroom is the best system but there are some who don't agree. LR uses a database to enable you to search for the picture you want. You have to work with the database to put all the appropriate data in there for the photos so you have a basis for a search. When you import your pictures into LR you can place tags (keywords) on the whole batch you are importing. When you go through them you can add or subtract the tags. Tagging is an art you have to develop. The tags have to be descriptive and complete enough to enable a search for several different characteristics of the photo. For example I generally put an event name on the batch of photos I import. Then if there are people in the photos I put the names of the people in the set of tags but only for the photos containing those people. The location also goes there (if not in the event name). You can also search by metadata, so the date and time will be in there automatically (from the EXIF data in the image file) as well as the camera name, ISO, shutter speed, aperture, lens focal length, and a lot of other stuff. It is a pain to sit there adding tags to your pictures, but it pays off in the long run.
The people who don't like LR seem to not like it on the basis of the database. They're probably not used to using a database and don't know how to use its power. If you have been searching for things by file name for many years, the LR database is different enough that you will have to get used to it. It's worth the effort.
Photoshop is another high-power image editing program. It has a lot of capability, so it takes a lot of time to master it. Again, it's worth it, and I should point out that you can derive substantial benefit from PS with only some basic knowledge of how it works. You don't have to be a grand master. The combination of LR and PS is really useful. LR is pretty much a basic editor with a few pixel-level tools. PS is a pixel-level editor. You can start with your images in LR and if the LR editor doesn't hack it you can send it right from Lightroom to Photoshop and edit it there. When you're done, you save your work (save, not "save as") and Photoshop sends the edited image back to Lightroom so it becomes an additional image in your database and you can continue working on it in LR.
Lightroom can be bought as a stand-alone program. Not so Photoshop, which is part of the "creative cloud" (CC) system, which you license instead of buying. The cost is about $10/month for a package containing both LR and PS. For some reason this honks off a lot of people. $10/month is really a small part of my photography budget. I guess they think that buying a program is a one-time expense. In reality, programs are updated at regular (or random) intervals and there's generally a cost to the update. In the CC system the updates are included in the $10/month.
Photoshop Elements is sometimes touted as "Photoshop Lite". It can be a stand-alone program but it has limitations. The major limitation is that the editing is done in 8-bit image space whereas Photoshop uses 16-bit. Since most cameras will do 12-14 bit depth, PSE looses some depth in the image rendering.
Another thing I would recommend is that you change your file names on download. My old P&S named the files something like "PA240001". Newer Nikon DSLRs name the files something like IMG_0001 or NEF_0001 (raw image files). There are only 4 digits in that name, which is based on the ancient 8.3 computer naming format. Computers will now handle longer file names so you should use them. You can make the name describe the subject of the shoot. Meaningful names are really useful because you can search them using the regular computer file search system. You don't have to go into Lightroom to do a search. I think this is really important because although I'm fairly proficient in using the LR search, someone coming after me may not know how to use LR. Meaningful file names makes it possible to find things if you only know basic things about computers. This is important for my family pictures. Probably not as important for my other pictures.
Lightroom can change the file name to your specifications on import. But before I found that it could do that I got another program, Downloader Pro that does the file renaming. I think it's better than LR in doing that because it can automatically send the image files to the right location on my hard drive. I have a more complete description of this at
http://www.uglyhedgehog.com/user-page?upnum=1595.
And I have a description of my LR workflow at
http://www.uglyhedgehog.com/user-page?upnum=1584Other things you should consider as a postprocessing newbie:
Regular backups. If you use Lightroom, it can be set to back up the catalog (database) on a regular basis
but it does not back up your photos. You have to do that separately.
Don't edit the photos on the camera card. Download the image files to your hard drive and work on them there. Then they're there the next time you need them. You don't have to have the card plugged into the computer, and you can reformat the card for further use.
Don't take anyone's description of their favorite program as "the only way to go". That is something you will have to find out for yourself. LR/PS is used by a lot of people (I would say for good reason), but there are other programs out there. Most of them have 30-day trial periods so you can try them out without having to spend a pile of money. The only problem there is that 30 days is really a limited time to try to compare several different programs, particularly since most people don't have unlimited time to do the comparison work. Minor things like work and family tend to get in the way.