From my experience, these are the best places for an amateur like me to shoot at the Indianapolis road course. In these locations, you don't need a special pass, and (with the exception of the front straight from Tower Terrace) you won't be shooting through any fences.

Pit lane/garagesPit lane/Garages
On the Wednesday morning before the inaugural USGP in 2000, I was poking around the back of the pit area, watching through the fence as one tire dolly after another was wheeled by, when a woman wearing a Speedway shirt said to me, "Go on in if you want to." I almost pinched myself! I walked past tire trucks, hospitality tents and still-covered containers of team equipment right into the pit lane! For the rest of the day, with only a break for lunch at the lovely Museum cafe, I wandered through the pits, chatted with early-arriving team members, walked the entire circuit and watched the Jaguar team practice pit stops. The following year, things were much less casual, but I still was able to spend 2-3 hours in the pits with all that exotic hardware.
Rooftop - Turn 13Rooftop - Turn 13
Absolutely the most outstandingly unique (as far as I know) aspect of the USGP at IMS for shooting pictures was a General Admission ticket that would get you into any grandstand on Friday and Saturday. On race day, the rooftop seats in Turn 13 (Turn 1 of the oval circuit) are among the most coveted in the world for any event. If you're willing to get there a half hour to an hour before a session, you could be in the catbird's seat.
Grandstand J - Turn 2Grandstand J - Turn 2
In much of the infield section of the circuit, fences are a major issue. From the top half of Grandstand J in Turn 4 of the oval circuit, you can easily shoot over them, though a fairly long lens (300-600mm) helps a lot. Since you are facing ESE from there, this location generally works better in the afternoon, especially on a sunny day.
Grandstand J - Turn 1Grandstand J - Turn 1
From just the right spot in Grandstand J, you can look over the fence, with the entrance of Turn 1 in the foreground, back up the front straight. It's not a great look at individual cars, but not bad for an atmosphere shot.
Viewing mound - Turn 9Viewing mound - Turn 9
The grass-covered infield viewing mounds, adjacent to the golf course, provide a few opportunities to view the cars up close in the slowest sections of the circuit. In this particular spot, there is actually no fence to look over, so expect a crowd on the side of the hill facing the gap. With patience, you can get yourself a clear view, make some great images, then move on and give someone else a chance.
Golf course - Turn 11Golf course - Turn 11
Just a few yards from one of the golf course tees, you can watch the cars exit the excruciatingly slow 8-9-10 complex of "Mickey Mouse" turns and begin to accelerate back onto the oval. You get a great three-quarters rear view as the cars exit Turn 11.
Corner 10 TerraceCorner 10 Terrace
The temporary grandstand at the exit of Turn 10 provides a great view of Turns 9 and 10, but I believe the best shooting spot is still in front of the stands at the fence looking through a hole cut for the professional photographers. A few feet makes all the difference there but everyone generally takes turns at the opening, and since only ticketholders for that grandstand have access to the fence now, there should be plenty of opportunity for a great look at the very slow apex and exit of Turn 10. Watch for cars to lift the inside front tire off the track here.
Tower Terrace - Main StraightTower Terrace - Main Straight
From the Tower Terrace, inside the circuit on the front straight, it is impossible to get an unobstructed view of the track, but the cars are going fast enough that the fence all but disappears in a pan shot taken at top speed.
Corner 6 TerraceCorner 6 Terrace
The top row of the grandstand outside Turn 6 allows you to follow the cars all the way through the medium-speed left-hand turn toward Turn 7 and onto the infield straight on Hulman Boulevard. As with the rest of the course, the track is completely flat here, but I often like to tilt the camera to add a bit of diagonal movement to the picture.
Pedestrian Bridge - Infield StraightPedestrian Bridge - Infield Straight
Strictly speaking, you cannot stand on the steps of the pedestrian bridge near the end of the infield straight. There is always someone doing it, however, and you can get a pretty decent view of whoever happens to come along before the event staff chase you away.
North Vista - Turn 4North Vista - Turn 4
Though it's a bit far from the road course, the North Vista provides a tremendous view of a long stretch of the infield from Turn 1 through Turn 6. My favorite shot from here is the cars entering Turn 4, the slowest one in this section.
Rooftop Stands - Main StraightRooftop Stands - Main Straight
The rooftop stands along the Main Straight, between Turn 13 and the Start-Finish line, provide a fabulous view across the track to the pit lane and team garages. Extra time spent staking out a good seat there for the qualifying session on Saturday is well spent.
In the Paddock!In the Paddock!
The Holy of Holies — the Formula One Paddock. It's Friday morning, an hour before the first practice session of the weekend, and I am among the glitterati in my jean shorts and windbreaker. You need a special pass to get in, of course, but if you have a friend who is willing and able to borrow one for you like I did (Thanks, Jeff!), you can wander among the drivers, team principals, and hangers-on while they eat, relax and primp. Behave yourself, show some discretion, and don't gawk!
Outside Turn 4Outside Turn 4
You have to be willing to arrive early to stake out a spot here because there is one gap in the fence that everyone will want to shoot through when the cars arrive. You should have plenty of time to get the shots you want and still be able to share with the others during a full session.
Northwest VistaNorthwest Vista
Perhaps the best place to view the first few corners of the race is the Northwest Vista. It's not an extreme closeup, but the cars come right at you as they exit Turn 2 and you can follow them all the way from Turn 1 through Turn 6.