...what are the relatively "nice" areas? The types of place where when houses in the neighborhood go on sale, they never make it into the MLS systems cause there is such a high demand? We have done the "new house" thing during our last 5 moves over the last 14 years and we vowed to each other that his time, we would move into an established "classic" neighborhood - we are looking to spend up to $1.0M...
I don't know if you'll find anywhere where homes are selling quite that fast, but there are quite a few neighborhoods that will no doubt be right up your alley.
The first place I would suggest is the Meridian Hills area. This area is located on the north side of the city between 42nd and 86th Streets. This is the "old money" part of the city where you'll find probably the most expensive homes inside city limits. One of the wonderful things about Indy, however, is that housing is super affordable here...I don't know much about the housing market for homes as expensive as what you're looking for, but I do know that you will get
a lot of house for your $1M in Indianapolis. The other advantage of being in the Meridian Hills area will be its proximity to the International School.
Continuing north, you'll find Hamilton County and the towns of Carmel, Fishers and Noblesville. Carmel is probably the most established of these suburbs, and is also the wealthiest. Geographically, it is situated due north of the city and extends slightly to the east.
The public schools in Carmel are nationally acclaimed and the facilities are top notch (take it from an alum

), so if you're considering public schools at all, I would start there.
Generally speaking, Carmel is divided in two by Keystone Avenue. The east end is the older, more established side, but is also less wealthy. The west end is home to some incredible mansions, but has been very aggressively developed in the last decade.
Fishers is another suburb, situated on the northeast side. My wife and I currently live there and absolutely love it. In general, Fishers is less wealthy than Carmel, and has absolutely exploded with growth in the last decade. This may make it a poor candidate based on your post. I would still check it out for two reasons: first, it is incredibly convenient due to its proximity to I-69. This major freeway leads directly into the I-465 beltway and into downtown. We go downtown a lot, and love getting there in 20 minutes because of I-69. The other reason to consider Fishers is the Geist area. This part of the town is on the Geist reservoir and has many beautiful lakefront homes in the $1M range.
The Noblesville area is north of Carmel and Fishers in the outer ring of suburbs. This area is the least wealthy of the three towns I've mentioned, but also has a reservoir with lakefront homes that are generally less expensive than Geist due to their distance from downtown. You might also like the Lochaven neighborhood, but you should know it is relatively new (broke ground 2002ish), and the Sagamore community, which is on a golf course.
One other suggestion: there are several older homes being re-developed in the Old Northside neighborhood immediately north of downtown. I would check out Delaware Street in particular. You should know that this area was just re-done recently, though, and until then was very ghetto and dangerous.
Good luck, and welcome to Indianapolis!