The aims of sauteeing, frying and browning onions, garlic, carrots or any other vegetable are to soften and/or add flavour to the vegetable by triggering browning reactions. Flavour can also be enhanced by careful choice of frying/sauteeing liquid.
Success with these techniques is partly a matter of experience; knowing what heat to use, how much water to put in so that you can brown your veg before you've boiled them to death and so on. These things will depend on what pan you use and what veg, and how you cut them, and so on and so forth. It gets easier with practise.
I first learned about fatfree cooking in about 1996 from Sue Kreitzmann, in her Slim Cuisine series and her later books such as The Complete Low-Fat Cookbook. I've also had lots of useful hints and tips from posters to the Fatfree Vegetarian mailing list.
A good nonstick frying pan is your best bet for sauteeing vegetables or for crisping up veggie burgers or tofu. Note that you shouldn't use a nonstick pan for the browning technique I describe below, though. A wok is great for low-fat stirfrying in a little oil, but I don't recommend using a wok for completely fatfree cooking.
I generally saute in some kind of flavoured liquid; the aim is to finish up with the vegetable softened and its own flavours developed, but with a film of flavour from the reduced sauteeing liquid remaining on the surface, or possibly absorbed into the vegetable (if it's a vegetable that will do that).
Put the chopped vegetables in a pan and add a little vegetable stock (yes, you can use stock cubes or powder if you want). Bring to the boil, reduce heat to a brisk simmer and saute! If it looks as though it's going to boil dry, add a bit more hot water and continue - it's useful to keep a just-boiled kettle handy for this purpose. You generally want to end up with all the liquid evaporated from the pan; even if you are going to add more liquid afterwards, if you didn't want to get the effect of the sauteed ingredients having individual flavour, you may as well have put all the liquid in at the start and boiled it.
You can use other liquids if you wish; such as red or white wine, or even apple juice for a slightly sweet flavour. Prune juice is a good substitute for red wine.
This is slightly trickier than sauteeing, but not by much. This technique will not work in a non-stick pan! Well, it may, but you'll damage your pan. You'll also need to have a jug of water handy.
Start off as for sauteeing, but let it boil dry. Stir constantly as it gets drier and the onions start to stick. A brown deposit will form on the bottom of the pan; scrape as much of this up as possible. Before it starts to go black, pour in a splash of water and use this to "deglaze" the pan - ie, dissolve the brown deposit and scrape it up.
The brown-ness will have transferred itself to the onions. Repeat this process as many times as you like. If you do it long enough you will get caramelised onions, a sweet, toasty-smelling collapsed mess. It tastes wonderful!