The most important point here is probably that you need to understand that there will be distortion in your images and it's likely that your source material wasn't carefully drawnin a particular projection. If it's just the traditional "flat map" that assumes that the world is truly flat, then you'll need to decide which is more important: relative shape or relative size. If you can go for some significant compromises in shape and area for large continents, then there are a couple of physical methods that should work nicely.
First, get a blank globe. There are discussions about getting a blank globe here on the forums, but the simplest way is to visit your local thrift store to buy a used globe and the hardware store for a can of white spray paint. The most basic method is just freehanding the shapes directly on the globe; that might be good enough. If you have a projector of some sort, then you can physically project the images onto your globe and trace them directly on the globe. Both of these techniques require a little artistic ability on your part.
Next on the list is to do as Hai-Etlik says, but there are a number of programs out there that don't require that you master GIS concepts. You'll need to understand that projections cause distortion, but other than that there's not much to do except decide where the continents go and work from there. I recommend using the Equirectangular projection for your intermediate projection because it's also the basic UV projection used in many 3D modeling program. If you can paste your images onto a sphere in a 3D modeling program, you can probably get it to generate what is effectively an equirectangular projection (and that is in fact method 3: use a 3D modeling program and use an orthographic projection to push the images down onto the virtual globe).
Long ago, I wrote a program called ReprojectImage ( http://www.ridgenet.net/~jslayton/ReprojectImage.zip ) that will let you take an image and drag a grid over it and see the results on an Equirectangular projection right next to it. Not perfectly accurate, but it's a pretty simple thing to operate. Reproject each of the images and then use something like Photoshop to assemble them and you'll have your equirectangular globe. Programs like Hugin will also do the same sort of thing ( How to create a little planet using Hugin | Ultrawide shows how to make a panorama, but the idea is the same).
I also wrote a thing for ProFantasy called Fractal Terrains that will ingest (among other things) an Equirectangular image and can output (among other things) an image in other projections. The basic package includes a number of projections, including interrupted sinusoidal and interrupted stereographic. You can define your own if you've a mind to.
There are many other tools out there that will do this same sort of thing; G.Projector ( http://www.giss.nasa.gov/tools/gprojector/ ) is a popular one.
Once you have the gores, then it's just a paste job. I recommend making separate endcaps for the globe to avoid having to fit lots of tiny slivers at the poles. Alternately, use a daisy-type projection ( Maps and Globes | USGS Astrogeology Science Center shows an example) and paste away from that. I keep meaning to teach Fractal Terrains that projection, but it keeps falling through the cracks in the too many things I have to do each year...
Anyhow, Hai-Etlik is a GIS person and naturally gravitates in that direction, with a focus on precision and accuracy. Most of the folks hereabouts gravitate more in the Photoshop and GIMP direction, with a focus on getting things to look good enough. I'm not saying that GIS things are bad, but most folks don't have the technical background or willingness to study long enough to get that background. For those folks, I've aimed more at simpler tools.
The way to preserve shape at the poles is to use a polar aspect of a projection. The attachment shows what a polar stereographic to equirectangular conversion might look like in the ReprojectImage program described above.