I'd recommend Natural Earth for this sort of data. It comes as shapefiles which can be read by any GIS system (e.g. QGIS)
Each country's record has a lot of associated data, this includes which continent it's in, so for example you could colourize each country by continent. You can also split layers out by field value, using a plugin. That'll give you your multiple layers.
There are a number of different scales; the 10m set is more detailed so more useful if you intend to zoom in; the 110m set is less detailed.
In QGIS you could export the resulting map (using print composer) to an image (png), svg or pdf. Or just save what you see on-screen if you don't need a high resolution image,
These sets of data are in WGS84, but QGIS will reproject to any of the projections you were after (note that equirectangular is called Plat Caree in the list of supported projections).
If you decide to try out QGIS, I'd recommend that you also download the graticules data set from NaturalEarth. This includes a WGS84 bounding box, and I find it's helpful to clip the countries layer to the bounding box when using certain projections, as you sometimes get odd rendering artifacts when things cross the 180 degree antimeridian.
Here's a quick example done in QGIS... colourized by continent, projected on-the-fly to Gall Stereographic