The armhole . . .
probably isn't something you want to touch. Most armholes are cut plenty big and are usually too large. Actually, that's why you can't reach for anything on the top shelf without pulling at your jacket. Modern armholes are cut for the large, so that no one complains of a small armhole. But before you go campaigning against ham-fisted people who need large armholes, cutting smaller armholes that can be tailored larger would create another problem: fitting the sleeves in. While an armhole can be cut larger, a sleeve cannot. Switch to custom jackets or shop entirely at estate sales for vintage jackets (armholes used to be cut smaller).
Shirt armholes can be made smaller, but only if you also want to taper the sleeve and torso. That lovely flat felled under arm and side seam, a hallmark of menswear, is one of the last seams to be sewn, thus making it fairly easy to resew.
Sleeves: Tailoring the sleeves can surprisingly tricky. Shortening lengths seems like it would be a simple affair: just cut off the excess at the bottom. This works wonderfully well at most hems - pants and skirts. But cuffs have wonderful things like vents (or fake vents) or plackets which allow the wearer to unbutton them and fold them up.
Jackets cuffs can usually be shortened a couple inches. The buttons, regardless of their functionality, are usually placed a couple inches back from the cuff. Your tailor should be able to shorten a jacket sleeve to half an inch below those buttons. Your tailor should also be able to lengthen a jacket sleeve up to three quarters of an inch.
If you need to shorten a sleeve more than that, or if you don't want to disturb your button placement or fuss with your surgeon's cuff, you can have a sleeve shortened from the armhole. Fair warning: this is a much more involved process and will be more expensive. This involves taking off the sleeves completely, trimming the excess, and reattaching the sleeve and its lining.
On surgeon's cuffs: they're awesome. They allow you to unbutton your jacket sleeves and fold them up when the weather is warm or when you feel like looking especially cool. They were once a hallmark of quality bespoke suiting and are always done on custom suits. Only recently have some ready-to-wear manufacturers started putting surgeon's cuffs on their jackets. Now, getting functional buttons on your previously un-surgeon-cuffed jacket is a tricky matter. You may have heard that only better tailors can do this. But you shouldn't count your tailor out if they won't make surgeon's cuffs on your jacket. The problem is that most contemporary manufacturers don't include enough seam allowance to make the vents for surgeon's cuffs and as amazing as your tailor is, they can't make something out of what isn't there.
Shortening shirt sleeves has a similar difficulty with the placket. It allows the wearer to unbutton it and roll up the sleeves. So, shortening the sleeve from the cuff shortens the placket. So, you're constrained by the lowest button of the placket. If you arms are very slim you may be able to shorten the sleeve past that button, but you'll have a very short placket. Shirts can hypothetically be shortened from the armhole like a jacket, but HRD wouldn't expect it to be worth the price.
Sleeves can also be taken in in circumference to some extent. Jacket sleeves can be tapered along the back seam. A tailor can't change the circumference much at the bicep, as it has to fit into the armhole, but can taper it down to the wrist. Shirt sleeves can be tailored along the underarm seam and the armhole can possibly be raised.
If your jacket or shirt is tight in the arms, there is less that you can do. A jacket sleeve can be tailored a quarter inch, possible a half inch, larger, but mostly farther down the arm than the bicep. Shirt sleeves can't be made larger.