First, there is almost never one "correct position". There are usually several different positions to take, all of which are legal and safe. However, there are always positions that are either illegal, unsafe, or both (illegal and unsafe often go hand in hand).
Having said that, there are two basic approaches here. When the right lane becomes a right turn only lane, you are approaching an intersection. One way to negotiate it is as a pedestrian: slow down, keep to the right, stop at the intersection, look back and make sure no right turning traffic going to cut you off, then proceed across the intersection, slowly, and ever-vigilant. If the intersection has traffic lights, then treat them per the pedestrian rules (wait for the walk signal, etc.).
The other way is to treat it vehicularly. How would a driver of some other slow moving vehicle go through this section? Say a fully loaded cement truck, a loaded moving van, a farm tractor, an aged Vespa, or a bull dozer? Where would they position themselves? The answer to that question is where the vehicular cyclist would ride. Note that this might mean staying in the right turn only lane for a while (if it's a long turn lane, there is lots of through traffic, and no turning traffic), before merging left into the through vehicular lane. It depends on the presence of other traffic.
The key principle here is: "use the rightmost lane that serves your destination". Another key factor to consider is whether that lane is wide enough to be safely shared side by side by bike and car. Usually that means 14 feet or wider. Anything much narrower than 14 feet, and you should be "centerish" - somewhere between the left and right tire track of the lane, exactly where depending on various factors and conditions.
If you're not comfortable riding it vehicularly, then take the pedestrian route I described first, at least for now. In the mean time, take some classes (see bikeleague.org) or do some more research on the topic. You might want to start with the "every lane is a bike lane" link in my signature. Good luck!