What type of vegetable oil should I use for tempura?
In my experience at expensive tempura restaurants, sesame oil is almost always used at least as part of the blend, but my understanding is that it is not the roasted kind, which has far too low a smoke point. I've been to a restaurant in Izu, now closed, where it was obvious because we were seated right in front of the fryer and the smell was quite noticeable, and the fancy place I went to in Tokyo was apparently using sesame oil as well, but they had more effective ventilation, so it was mostly obvious from the flavorfulness of the results. I've seen unroasted sesame oil in middle-eastern markets and I know some companies sell it in the US, so it is obtainable, and I've seen it at about $40/gallon on Amazon. I also know from experience that a blend of the roasted sesame oil and neutral, high-smoke-point oil will have a higher smoke point than just straight roasted sesame oil, and will add flavor, so it's at least possible that some people do that.

At home and at inexpensive restaurants, other vegetable oils are frequently used, and there's a simple tempura set that you can buy as a gift pack that I've seen used as a winter gift. This is usually cottonseed oil or a blend. In some regions, an ambiguous blend called "salad oil" is commonly used. It is not unheard of to use tea seed oil (camellia), which is increasingly available in the US at Asian markets (when I ran a specialty import business, I used to sell sell some that a Chinese tea importer was purchasing until he shifted his focus to larger packages suitable for restaurant use).

The Japanese entry for tenpura on Wikipedia confirms that a custom blend of sesame, cottonseed and other oils is often used at tempura shops. The Tokyo style and Kansai style is a bit different, with many Tokyo shops serving a darker version thanks in part to sesame oil and egg, and Kansai region versions leaning a bit whiter thanks either to using neutral vegetable oils or lack of egg.

I've used canola oil but it tends to leave an unpleasant aroma in the air, especially after the second use. If I were in the US and on a budget, I'd probably choose peanut oil, but it will taste different than most of what I've seen in Japan. I doubt that I've ever encountered peanut oil-fried tempura in Japan, though there's a chance I just didn't notice.

In practice, I've found that the most important thing to get good results from homemade tempura is to prevent overloading the fryer, because the temperature will drop too quickly. This is true for other kinds of fried goods as well, but is far more important for tempura because the coating should be fairly thin. As an unpracticed amateur, I've found that using a modest amount of katakuriko or cornstarch in the flour blend helps produce crisper results.

It's fairly difficult to get an assortment of different items to be ready at the same time, so consider making it in "courses" if you want flawless results, or make only one or two items at a time. Tempura restaurants tend to have wide, slightly shallow fryers, whereas most fryers meant for western deep-frying tend to have several stations of deep pots, since it's ok in many cases for things to be submerged, like when making french fries. It's quite unusual for tempura to be completely submerged.

Edited to add:
Assuming you're coating with a batter (there are some types that are dusted with flour or katakuriko rather than a full batter), I get far better results by keeping the batter (or the flour) very cold.

Many years later edit, since I live in Japan now and occasionally buy prepared "tempura flour":
I just wanted to note that, looking at the ingredients in tempura flour sold in Japanese supermarkets for home cooks wanting to make tempura here, baking powder is typically in the mix. I have added it occasionally with plain flour in the US depending on how important crisp results were to me, but if your primary issue is textural it may be worth a try. I don't think it's as much baking powder as "self-rising flour" would contain, but I'm not sure. I will say that in a pinch, when I was low on regular flour, I used tempura flour in scones and got acceptable results, so it may be "just enough" to cause some carbon dioxide explosions that produce a leavening effect.