Texture format can and does have an effect on how the texture looks like and how fast they are rendered.
PCX textures are indexed to use 256 colours and they pretty much suck because they still use as much memory as a 24-bit TGA of the same resolution. Additionally, the indexing eats colour depth.
TGA files are bloody big and thus take a lt of HD space and can take a lot of memory, but not any more than a PCX or JPG
of the same image size. But TGA's also have the best image quality especially in effect textures that use a lot of smooth gradients.
DDS files are for most purposes the absolut best texture format. Especially for large ship textures with few gradients. The compression is usually rather good, but the main advantage is AFAIK that the GPU can handle them directly in their compressed form, which means that the memory load will be something like 1/4 of uncompressed images (depending on what there is in the image, obviously. Also, as was already said, if there's no need for transparency, basic texture and glowmaps should be DXT1a compressed (and also some effects that don't use alpha blending properly yet, like the beam textures). If the purpose requires alpha channel, then use DXT5 compression... or DXT3, if you deem it good enough, basically they are the same but DXT3 has less depth in alpha channel.
JPG... just don't use jpg textures. They might be small to transfer, but the compression artefacts are far worse than with DDS textures, and they use the same amount of RAM as uncompressed textures.
Obviously there is no way any format can magically make crap textures look any better, but it does have an effect on how good a texture can look. There's really no contest between 8-bit and 24-bit colour.