We will start with the upper leg. First, hide everything except for the leg. To do this select everything except for the leg (in object mode) and press Ctrl+h. On Macs it is command + h. Alternatively, you can find it in Display -> Hide -> Hide Selection. Once this is hidden, create a sphere. Set the Subdivision Axis to 10 and the Subdivisions Height to 6. The Subdivision Axis on the sphere must be set to whatever the Subdivision Axis was set to on the cylinder since each vertex in the middle of the sphere must have a corresponding vertex in the cylinder. Orient it so that a pole of the sphere is facing the cylinder and scale it so that it fits on top of it.
Next delete the bottom half of the sphere by going into subobject mode, then face selection (as we did when creating the hair), select the bottom faces, and press Delete. While you are in subobject face mode, delete the top faces on the cylinder by dragging a box around the top half of it and then pressing Delete.
Next go into object mode and select both the top half of the sphere and the top half of the leg. Go to Mesh -> Combine. This will combine those 2 pieces of geometry into one.
Go into subobject vertex mode again and click on the snap to point button . This will only allow you to move vertices to other vertices. Select each vertex on the sphere and move it to its corresponding vertex on the cylinder. A good tool to use here is the arrow keys. If you press the arrow keys you can move between vertices. The end result should look like this. Notice the sphere seamlessly merges into the cylinder.
Although they are in the same place, the vertices on the sphere are still separate from the vertices on the cylinder. To merge the two sets of vertices select all of the vertices at the bottom of the sphere and top of the cylinder by dragging a box around them and then click on the box next to Edit Mesh -> Merge. Make sure the Threshhold is set to 0.0001 and then click Merge. Now turn off the Snap to Point button and repeat this process of attaching a sphere to the end of a cylinder with all of the joints.
To unhide the objects last hidden, press Ctrl + Shift + h (on Macs it is command + shift + h). To unhide any hidden object that may not have been the last thing hidden, go to Window -> Outliner. This lists all of the geometry created so far. Anything hidden will appear in blue. To unhide something, select it in the Outliner and then press Shift+h. You can also select multiple objects in the Outliner by shift + clicking.
Once all of the joints have been fixed, the result should look like this:
Notice I put spheres at the beginning and end of the lower leg and lower arm because they fit better than they would on the hand and foot. Also note that the sphere at the bottom of the neck needed 6 Subdivision Axis and not 10 because there were only 6 in the neck cylinder.
Now for the final step, you must mirror the arm and leg. Before you mirror, make sure the pivot points are on the middle axis because an object will be mirrored about its pivot point. To do this press the Insert key (or Home key on Macs) to move the pivot point of an object. Then click on the snap to grid button . This will snap the pivot point to grid lines. Since you will only be mirroring the arm and leg, you only need to do this for the parts of the arm and leg. Select the top part of the arm and click and drag on the horizontal line coming out of the circle of the pivot point. Drag that to the middle grid line. The Y coordinate of the pivot points does not matter, just the X coordinate must be set to 0 (i.e. on the vertical grid line). To exit out of Pivot mode, press w to bring the translate gizmo back.
To mirror these, select all of them and go to Modify -> Freeze Transformations. Then click the option box next to Edit -> Duplicate Special. Set the Scale to be -1 in the x direction, and make sure the Copy button is checked. Then click Duplicate Special.
After this you will need to flip the normals on the duplicated parts. Normals are lines perpendicular to each face that tell the computer which side is the outside. Since the mirroring made the arm and leg have a negative scale value, their normals are flipped. To flip them again, while they are still selected go to Normals -> Reverse.
You may want to tweak the vertices on your model after this is done. Now would also be a good time to start naming the parts on the model.
To name the parts on your model, open up the outliner (Window -> Outliner). You may notice some items got into groups when you combined them. A group is signified by the icon, and geometry is signified by the icon in the outliner.
You will want the outliner to list only geometry pieces with no groups. To do this first select everything and go to Edit -> Delete by Type -> History. This will delete the history on the model which will in turn get rid of some unwanted things in the Outliner. It is a good idea to delete the history on the model often because when the history gets too big, Maya runs slow. When deleting the history make sure you are in object mode and not subobject mode.
If you still have some groups after deleting the history, expand the group, drag the geometry outside of the group by clicking and dragging the middle mouse button, and then select and delete the empty group.
To rename a piece of geometry, select it in the outliner, double click on it in the outliner, and then type in the correct name. You will know what to name it because the geometry will be highlighted in the viewports. Here is what they should be named: