It depends on several factors. I replaced a window on our 1995 built two story two years ago. We have aluminum siding and I realized the window was not flashed properly when it was installed. I needed to remove the siding from about half the height of the house, remove the rotting window and frame, repair/replace the damaged framing/sheathing near by, install the new window, flash it properly, replace the siding. That was more complicated than the scenario you have, but you have more of them.
If you don't feel drafts around your windows, the framing is probably sealed adequately. You can buy insert windows which only replace the window glass and frame and not the sashes. These can be relatively inexpensive. I think Marvin makes a decent product for these, but others probably do as well (Anderson, Pella, etc.). You can also buy replacement windows which replace the sash and the whole assembly, but typically you need to have wood/hardboard siding to make this work or the original window doesn't have built in trim.
As far as buying good replacement windows, stay away from the big box stores. I'm not sure where in Minnesota you live (I'm in Minneapolis), but a local lumberyard is a better bet. I bought my replacement Marvin window from Scherer Brothers in Hopkins as it was convenient to pick up on my way home from work. Hopefully that helps.
As far as feeling the sun through the window, that's normal. It's the radiation part of heat transfer. Double/triple pane is addressing the convective part of heat transfer and increasing the thermal resistance of the window. You could buy a window with a coating to reject more solar radiation, but it's not really needed in Minnesota (maybe Arizona). If you feel drafts, that's a bigger issue as you're exchanging air with the outside in an uncontrolled way (you do need some building ventilation).