Mostly correct... but not about the resolution, though it is quite the dastardly subject, so the confusion is understandable.
"Resolution", as referred to by imaging/photo apps like JASC or Photoshop, is meant for sending files to print. The purpose of increasing image resolution is to provide more "dots per inch" of image information to the sheet of paper going through a printer. The more dots you can cram in an inch on a sheet of paper, the more detailed the image looks. However, your computer monitor (which has pixels instead of dots) can not cram more pixels on an inch of itself. The monitor has a fixed amount of pixels.
These image/photo apps make up for that issue by increasing the actual pixel dimensions of your image... let's say you wanted to create a 100x100 pixel square image for your website. With a "72dpi" resolution (standard screen resolution), your image would indeed be 100x100 pixels.
However, if you bumped up the resolution to, say, "300dpi", you'd see that your image has now changed dimensions... it's more like 400x400 pixels now. This works out fine in the end when you're printing, because the printer goes by the actual inch/cm dimensions of your image, and cares not for the pixels (it will however place "300 dots" of ink on every inch of its paper, making the image the same size as you originally specified, only more detailed)... but if you're creating an image for the web, you would end up with a 400x400 image instead of a 100x100.
Even if you changed the resolution to 300dpi and THEN changed the pixel dimensions themselves back down to 100x100, your image would not look any better than the original 72dpi version; again, your monitor cannot add pixels in between other pixels. It will render the same quality 100x100 image, whether you have it at 72dpi or 7200dpi.
In short- image resolution means absolutely nothing to PC monitors; don't change the resolution of the image unless you're making print graphics. The other advice given is all you need- use actual art programs to make large graphic text (like the Photoshop you supposedly don't have a need for), and if there are style options like "Crisp" or "Sharp", make sure you specify which one you want.