Each has their own advantages and disadvantages.
You will need to choose before you complete your new door installation.
Let's start with solid core...
what does it mean? The center of the door has some sort of wood or composite that is covered by the outer skin or veneer.
In a high end door, the wood material you see on the outside is actually solid all the way through.
So, if you purchased a solid cherry door it will be cherry wood through-out.
Be careful when you think your purchasing a solid wood door that you don't end up with a solid core veneer.
There is a difference.
The major advantage to solid core doors is they are more sound deadening than a hollow core door.
Another advantage is they are stronger.
They will take some abuse without much more than just scratches.
The disadvantage is the weight.
They are heavy and hard to handle all the way from the lumber yard to your house.
The weight of the door can cause some drywall cracking if the door gets slammed hard enough.
So if plan to have any door slamming competition with your spouse or one of your kids; you may not want a solid door.
Another disadvantage is with more wood, more warping can happen.
Take care to always apply a finish to all 6 edges (top, bottom, edges and both sides) of the door.
This will help keep the moisture content of the door more consistent.
Now hollow cores...
they normally have a honeycomb cardboard material in the center of the door.
I know it sounds cheesy but they are surprisingly very strong and are much lighter to handle.
Because there is less actual wood in the door there is usually less warping.
However, special care must be taken when applying the finish on your hollow door.
I have seen doors freshly painted and then stood on an angle to dry.
The door gets wet from the paint and becomes somewhat flexible.
It will dry and possibly keep the shape of when it was wet.
Always try to support the door as not to cause it to bow or twist while the finish is drying.
You must still be sure to finish all 6 edges.
The biggest advantage is the cost.
You can get very impressive looking doors for much less than solid wood doors.
Door manufacturers have made great strides in producing very realistic replicas.
The major disadvantage is probably its strength.
If impacted hard enough, you can pierce the veneer skin leaving a hole in the door.
Once you have decided which door style and type you want; check out some installation instructions.