This may not be much, if any, help, but I thought it was worth pointing out that tailoring something like a shirt (any shirt, much less a dress shirt) is a lot harder than hemming a pair of pants (and even doing a good job of that isn't as easy as one might think) so, frankly, unless you're willing to "invest" a lot of time and effort into this, it might (really) be more cost-effective to have your shirts made to order, especially if you're as picky as it sounds like you are about the fit. I don't know how prevalent this sort of thing is in less densely populated areas, but in big(ger) cities, there are often small outposts of overseas tailors that take your measurements and then make the shirts elsewhere (Hong Kong used to be a prime spot for that, though I imagine globalization-induced inflation has pushed much of it elsewhere these days). The less expensive shops' shirts obviously won't look like they were made to order at Paul Stuart (and you won't get label recognition, if you're into that), but while I never bothered to look into it for myself, the few people I've known I've been told they usually have a decent range of fabrics to choose from (at different prices, of course) and the ones I've seen have seemed at least as well made as good "department store" brands (by which I mean "designers' " off the shelf stuff.)
As far as altering existing shirts goes, consider the number of small parts/panels, and even greater number of seams, on a shirt. Taking them apart (without screwing them up in the process), altering them however slightly, and then getting them back in place more or less perfectly is no mean feat. And perfection counts for a lot. "Simple" clothes are a sort of thing that scream some thing's wrong when almost any thing about them isn't quite right, even if you can't put your finger on it immediately: even slightly non-symmetrical panels of fabric, slightly puckered, or not-perfectly-straight, seams, and/or (again, even slightly) uneven stitching will all make a what might've started out as s nice looking shirt look worse than a cheap one straight out of the package. And then things like shoulder panels, collars, and cuffs have multiple layers of fabric and/or interfacing, which are a whole ball of wax unto themselves. And all that is in addition to the basic talent and skill involved in designing and fitting together all those little pieces and seams in a way that produces the overall look and fit you're after. Trying to learn that from scratch, by trial and error no less, would be a long, not very entertaining, process...
As for sewing machines, I started looking into them with an eye to buying something in the price range you mention (just for household utility purposes like hemming sheets and bedspreads, altering window curtains, etc, rather than tailoring clothes) and even for that limited purpose eventually gave up (for the time being) in frustration. After reading countless product reviews and a lot of blog posts and articles on various sewing/crafting-enthusiast websites, the basic impression I came away with was that cheaper modern machines really just aren't very good. Like so many appliances, they've gotten a lot "fancier" as far as superficial features go (large numbers of automated stitch patterns, programmable stitch patterns, etc), but that's because that stuff is easily programmed into the relatively inexpensive computerized control circuitry, while the more expensive, but fundamentally more important, things like the bodies, motors, and overall design and construction have, not too surprisingly, become steadily "cheaper" (in the pejorative sense). Supposedly one can sometimes find good deals locally on used and refurbished machines, but I looked into that a bit, too, and was frankly unimpressed with the prices I saw. The less expensive machines were basically the same crappy models that are fairly cheap even new, and overall, the prices surprisingly (to me) weren't really discounted much compared to new. But presumably that's a YMMV thing, so it might be worth looking into, especially in geographical areas with a lower CoL (and commercial rents) than NYC...