I have heard that roasted vegetables are more tasty than steamed, but I have never been successful in roasting in small batches.
What is the best temperature to roast and for how long? Should I add olive oil? How much?
Preheat your oven to 425F (218 C). Squashes and root vegetables will take about 45 minutes at this temperature. Mushrooms, cauliflower, broccoli, and brussles sprouts will cook in about 25 minutes. Cut your vegetables into evenly sized pieces. Place in one layer on a baking sheet (parchment lined is good for browning and easier cleanup). Drizzle generously with olive oil. Season with salt and pepper. Roast. Turn the vegetables over halfway through the cook so that they brown evenly. This information, with pictures, can be found here.
Yes, roasting vegetables has more flavor than boiling or steaming, because the higher heat will cause browning, which creates additional flavorful compounds in the food. You also reduce the moisture in the vegetables, which will concentrate the flavors, and you don't risk washing away water-soluble flavors as happens with boiling/simmering.
But as this came from the parenting site, I don't know if you necessarily want this to be too flavorful, if you have picky eaters, or you're intending this to be pureed into baby food.
I don't know if there's a truly "best" way of roasting vegetables. If you're cooking from raw, you need to roast at a lower temperature, so that the inside is cooked through before the outside gets burned.
You can also adjust this by how you cut up for vegetables; the smaller they are, the faster they'll cook. But don't go for too small of a dice, or it's more finicky to cook. You're better off with thinner slabs to cook things quickly, although I'd also make it bite sized for the smallest mouth at the table.
I agreed with moscafj about oiling your vegetables, but I'd recommend that you mix the vegetables in a bowl before spreading it on a sheet pan. It's just easier to make sure that everything is well coated without making a mess. (and if you have children, this is a task that they can do). You can use a slotted spoon so you don't end up with an oil slick under the vegetables if you've added too much. I also take the opportunity to season the vegetables in the bowl; I typically add some herbs and a splash of vinegar along with the oil and a few pinches of salt.
Spread the vegetables out in a single layer, but you'll also want to make sure that they're not crowded -- there should be some space between then, not packed in tightly.
Because of the 'bite sized' issue, and firmer vegetables (carrots, winter squash, potatoes, sweet potatoes, etc.) take longer to cook, you may want to start some vegetables before others; start with them, then add peppers, onions, summer squash, etc. about half way through after you've flipped over the firmer vegetables. (exact time depends on how large they're cut up; check recipes online for a starting guideline).
And then there's another trick ... You can simmer potatoes and other firm starchy vegetables until they're just cooked through (but not soggy), drain them, put them in the bowl (or back in the pot), and knock them against each other before oiling and seasoning. This will rough up the outside, creating more surface area so it crisps up more. And they'll then cook in a similar time to the other vegetables.
You can also pre-heat the sheet pan, so you'll get a little bit of extra browning and possibly avoid needing to flip the vegetables when they're cut fairly small.
So, to summarize:
Try frying in a pan on your stove, or on your grill if you have one.
Chop/Slice/Dice your vegetables into pieces that seem appealing to you, toss with a bit of olive oil, and fry over medium-high heat.
Let the pan heat up first, then add the olive oil, then the vegetables. Toss in some kosher salt and some fresh ground pepper, and stir. Stir every-so-often to ensure nothing burns, but you do want to develop some brownness which is where a lot of flavor comes from.
After the vegetables have developed a little brownness, or become a little soft, you're done! Usually takes about 10-15 minutes, depending on what you're cooking.
You can even try drizzling with balsamic vinaigrette, or other coatings for an extra tasty side.