Please forgive the incredible length.
I had what I consider successful cataract surgery on both eyes last fall. I'll mention some of the issues/decisions I faced, but please understand that my eyes were extraordinarily bad, requiring very powerful progressive lenses and leading to up to 10 annual visits to doctors and opticians as I tried to get lenses that worked for all aspects of my life, and I knew going in I'd end up still needing glasses. Most patients will have a less complex experience, and possibly fewer decisions to make.
The patient is normally awake during the surgery. I was given something intravenously which may just have been a tranquilizer; I needed either more of it or a second drug because bright lights make me nauseated, and that had to be prevented. The visual effects during surgery were not unpleasant--a bit like looking into a kaleidoscope. My only issue was with the bright lights, something that isn't a problem for most people. I don't remember any pain other than from the lights, before the drugs kicked in.
The procedure MacMadame mentioned (one eye corrected for close-up use, the other for distance vision, intended to totally prevent the need for glasses) seems not be be used as often now as it used to be. I believe quite a lot of folks didn't adapt well to the results. This is something you can discuss with your surgeon and think about. I had always felt funny about that concept and was relieved that it didn't seem to be so much in favor as of last fall--one less decision for me. It's possible, though, that it would be worth considering by others (maybe especially for those with much better native vision than mine). In the end I ended up with somewhat different vision in my eyes; the final decision about what to do with the second eye was made after we had the results from the first eye.
I assume it is standard procedure to first meet with the surgeon to discuss the procedure and your wishes (try for perfect near vision or perfect distance vision, or something in between?), then have the measurements taken (this is non-invasive, via camera). The surgery will be later still, and you'll probably need a pre-op physical before that. There can be a long wait to get the initial appointment, and I assume also for the surgery on the first eye. There will then be at least a 3-week wait, I believe, before the surgery on the second eye. After each surgery you will have follow-up check-ups; I think 3 per eye is probably standard, but if you end up with (short-term) pressure build-up in an eye, as I did, you may be revisiting your surgeon daily for a few days. They will check the vision in the eye several times during the recovery period. You will have multiple drops to use for some time after surgery.
Cataracts can go from not noticeable by the patient to severely impacting vision rather quickly, so I wouldn't want to wait until I had really significant vision issues to start the process; doing that could mean months living with severely impaired and deteriorating vision.
Do you have significant astigmantism? If not, skip this paragraph. If you have significant astigmatism (as I did in one eye), you will probably have a choice of a regular lens implant or a special (toric) lens to correct some or all of that problem. I had to pay several hundred dollars out of pocket for that lens (the other costs were covered, mostly if not entirely by Medicare). The toric-lens bit is tricky. First, if you've got vision as bad as mine, the eye measurements previously mentioned may not be 100% accurate, so you may not end up with quite the intended vision. I was told to use warm compresses on my eyes twice a day in the period leading up to the measurement, because dry eyes can aggravate that problem. Second, regular corrective lenses come in different sizes; the toric lenses do not. My toric lens rotated a bit out of its intended position, so it isn't doing as good a job of correcting my astigmatism as would normally be the case; however, I only notice the problem if I just look through that eye--not something one normally does. In addition, my toric lens ended with a bit of a wrinkle in it, so I needed a quick follow-up laser treatment to sort of tack it down. That was quick, painless and done in the doctor's office. Fourth, the futzing around during the surgery trying to get the darn lens in the right position extended the length of the procedure, and I ended up with elevated pressure in that eye until I used special drops for a few days.
In addition to having a dominant hand, you probably have a dominant eye. In my case the dominant eye is the one that had slightly better vision and little astigmatism. (Maybe the dominance resulted from the difference in vision quality??) It's worth discussing with your surgeon whether there's an advantage to starting with the dominant vs. the non-dominant eye, or with the more problematic vs. the easier eye. My bad, non-dominant eye was operated on first, and I was glad of that since it didn't turn out quite as well as we had hoped (and we were just hoping for something like 20/100 vision). I popped the useless old left lens out of my glasses and was able to get around fine outside my apartment with my dominant eye + old lens overriding the corrected but far from perfect left eye for which I didn't yet have a new lens.
There will be multiple weeks mid-surgery when the operated-on first eye will be much better than it was before, but you won't be able to get a new contact lens immediately if you need one, so you may not be seeing as well out of that eye as you were before (pre-cataracts but with contact lens). That can be disorienting. I had to read with the left eye closed, because improvement the distance vision made my near version much worse. Since it felt so odd Based on how odd it seemed not be be able to read any printed material through the left eye without glasses, I asked to have the right/second eye corrected so that I could read through it, in a pinch, without glasses. This means that, without glasses, my distance vision is now worse in my right eye than in my left, the reverse of what I previously had. So I guess I ended up with a mild version of what MacMadame described, but since I still use glasses (graduated lenses) all the time, my brain is not having to process two massively different types of vision.
The results: I'm still dependent on progressive-lens glasses. My distance vision is noticeably better than it was before (even with glasses), I can get around safely (but couldn't drive) without my glasses if I have to and I no longer struggle to read things on a computer screen (though I display most text at 130% to make it easier on myself). My book-reading ability is not as good as it was prior to the cataracts. We're still tweaking the reading part of my prescription and figuring out how the glasses should sit on my face, because things are a bit off, especially if the type is 8-pt or smaller. This is frustrating, but I'm not having nearly as much problem getting workable glasses as I did before, because I no longer have such a massive difference between the corrections needed for near vision and distance vision.
I recommend discussing how much you sit in front of a computer screen vs. how much time you spend reading books with your surgeon so that he/she can try to optimize your vision for the activity most important to you. Keep in mind that how you spend your time may change after retirement (if you are not yet retired), but you'll be living with the results of the surgery for the rest of your life. (The computer-vs.-reading decision may be a non-issue for folks who don't have the dreadful vision I was dealing with.)
I had rather dry eyes before the surgery, though not to the point that I was using eye drops. Surgery may have made that a bit worse, but I'm still not using drops. The usual treatment (per multiple doctors) is: wash eyelids with baby shampoo when you bathe/shower, take fish oil capsules daily, and use warm compresses once a day. You can get a microwavable eye mask for this purpose at CVS or order them online.