Redshift means all the light emitted from what you looking at is shifted to longer wavelengths. So blue might appear green, green might appear red, red -> infrared etc.
Blueshift is the opposite: wavelength gets shorter.
Now here is the clincher: Quantum Mechanics. The quantum mechanical nature of the atom means that each of its kind will have a definite "fingerprint", i.e., emit/absorb light at very well defined wavelengths (colours). When you take the light from the star, or any luminous object, and split it up into its pure colour components (e.g., by passing the light through a prism), you get something that looks kind of like a barcode with dark and bright bands at specific colours (spectral lines). This "barcode" is composed of the fingerprints of the atoms that make up what you're looking at. These fingerprints are highly detailed and there is no mistaking one type of atom for another (or molecule for that matter).
Redshift shows up in a most convincing way when you look at these fingerprints, and the pattern is the same as that of hydrogen or helium atoms say, but it appears at the "wrong" colour: e.g., the blue lines are green, the green lines are red etc, but the location of the lines are the same with respect to each other. The only way this can happen is by Doppler shift with the source moving away from you. (Blueshift is the same thing except the colours and source move in the opposite direction.)
What & How-to:
So if you see a star through your telescope and it looks red, it does not give you enough information to decide what is going on. It could be a really old star running out of fuel. Or it could be a really bright young star moving away from you at a significant fraction of the speed of light. The way to tell which explanation is correct is by using light spectroscopy to show if the spectral lines are shifting, and if so, which way and how much.
Amateur astronomers can and do do this. It's not very hard but it's very nerdy. Much more nerdy than looking at the Moon, Saturn's rings or even hunting down faint remote galaxies or taking photos of these things. Lot less glamorous too (which is why it's so very nerdy). But it's not that hard to do. You need a diffraction grating (which acts kind of like a prism, splitting light into its composite colours) and a camera/sensor to match, and the usual astrophotography gizzmos. And a lot of dedication.
I've seen the spectral emission and absorption lines (bright and dark) from the Sun and other self-luminous objects through a spectrometer I made myself from readily available inexpensive components - total cost < $10. You can clearly see the fingerprints of hydrogen and helium in sunlight, of mercury and other nasties in fluorescent lights. Starlight is a lot dimmer though than the Sun or your energy-saving globe. So typically you'd need to capture the decomposed light on a sensor with long exposures to see what's really going on.