This is one of those questions that there are a lot of different opinions on. Personally I think that you should look at a few different things before you make a decision on what type of car would be best.
First of all the statisitics on teenager drivers in the US are not good. In the first six months of driving a driver between the ages of 16 to 19 are 11 times more likely to crash than a 40 to 50 year old. Motor vehicle crashes in the US are the leading cause of death of 15 to 20 year olds. Then you also have the statistic that a motor vehicle crash is 3 times more likely to occur after 9:00 pm.
There are a few factors that are behind these statistics. Teenager drivers are still developing their skills, they have a lower ability to detect a hazard, a low perception of risk, they tend to take more risks and have a higher statistic of not wearing seatbelts. Also teenagers with passengers have a 3.6 times higher risk of crashing than if they were driving alone.
Due to some of these statistics, safety should probably be the most important factor in the decision making of what type of vehicle to drive. Therefore I would caution you about "reconstructed title" vehicles. If a vehicle has been in a major crash the crumple points on the frame that protect the cabin of the vehicle have already been utilized. So the structure of the vehicle is in a compromised state, and that vehicle would not hold up as well in another crash. Before purchasing this type of car I would recommend taking it to a vehicle frame specialist to check out.
The other things you want to look at are:
Teach your new driver to budget about $800.00 per year for maintenance and repairs. On new cars you don't need that much, but if you save that yearly, when a 30,000 mile maintenance or new brakes or etc… come up they are always prepared. On a used car those funds will most likely be needed to maintain it properly.
As far as, Make or Model of a vehicle each person has their own preferences. Specialty cars, sport cars, and European cars, are more intricate and can be much more expensive to repair and maintain. Asian and Amercian made vehicles can cost less to repair and maintain, but I would double check with your insurance company on the statistics of the year make and model that you are interested in, as to how often they are vandalized and stolen.
So, before you or your teenager purchases a car, look at safety ratings, Car Fax Reports, check with your insurance company, and then find out what that vehicle requires for maintenance and how much those costs are. If you buy a vehicle used we recommend getting a Used Vehicle Inspection by a qualified repair facility so that you know how good of shape that vehicle is in.