Hello, MCsAngel. Welcome to the boards!
I'm a strict veg, so for me, no eggs means exactly that - no eggs. I used to eat them on occaision even though eggs are also on my "can't eat" list which meant that I shouldn't have been doing that, even though I did, anyway. My health was not taking very kindly to my allowances, so I had to come down hard on myself and say "no more eggs, Dee". It's hard, but not as bad as I thought. Sometimes the bread I get has eggs in them but I just keep in mind not to eat that much bread while we have "brand X" in the pantry. Plus, I keep a supply of pitas in the freeze for just those very times (saves me, for sure) and since I'm not much on fake meats (a taste thing, purely), I don't have to worry much about the ever-present egg whites contained within. Besides, I've recently discovered Gardenburger Riblets and IMO they're much better than a vegbrat any day of the week.
For baking, there are a number of things you can use as an egg sub. Ener-G replacement powder is good, I hear, but only if you're replacing up to 2 eggs - any more than that, and the recipe might not turn out well. Tofu, bananas, or applesauce is also good but again, it depends on what you're doing that will dictate which ingredient makes the better sub. If you have any question on what can replace what, you can always check out the vegan cook sites, like www.VegWeb.com/food
to see if what you're wanting to do will work. Our own "food and recipe" forum is also a great place to ask about subs since we have a lot of really amazingly talented homechefs that are willing and happy to help.
For eating out, there are some rules of thumb that I use. Bread or anythind made with or from bread will almost always have eggs as is most fresh made pastas. If they use dried pasta, you usually don't have to worry about it because it's rare that a dried pasta will contain eggs, but if they make it fresh, then you know that more and likely, that fresh will have eggs in it. Batters will more and likely have eggs in them as well but you can always ask the waitstaff if you're unsure (I personally just assume they do and order something else, instead). Sometimes, fried rices or certain soups will have eggs in them as well as any sealed thing (like an eggroll or stuffed wonton) because egg is used as "glue". I'm not sure about Olive Garden since I haven't eaten there in ages, but there are threads about what they have that are compatible to different lines of diet. That information would probably be best found in the "product reviews" forum and if not (because the thread was too old, or something), you can always ask.
That's all I can really think of, right now. Hope this all helps.