JPG's are 'lumps & bumps', or pixels.
When upscaling Jpegs, the image quality can go from LEGO to DUPLO, as the existing pixel is stretched. The best way way work with JPG's is to start with as high a DPI or PPI as possible, 300 DPI is always going to look better than a 60 PPI icon. Some graphics programs can 'smooth' (add more dots) to an image before scaling.
Photoshop, Photoshop lite/student, PaintShop Pro, GIMP, & CorelDraw/CorelPaint can do this.
Vectors are 'lines & degrees', so when you upscale, there shouldn't be any scaling issues.
What Don S says is good, but can you post an pics of what you are trying to do, and what programs you are using? MS Word, Publisher, even Paint can do a lot with pics & scaling, but not with perhaps the finesse of (expensive) other programs.
Edit: JPG/BMP to Vector programs do exist (Corel and Photoshop, and others) but are a whole other learning curve.
Good luck