I've used my EF 70-300 f/4-5.6 IS USM with a Tamron 1.4x and 2x however I am
not totally pleased with the results.  You should put it in manual focus
mode.  With the 1.4x and a bright high contrast scene, the autofocus
confirmation indicator works.  The IS works pretty well with the TC as well
but it is better on a tripod.

The main problem is that the resolving power doesn't increase that much.  It
does some, but not as much as you might expect and getting accurate focus
isn't easy.  So it isn't good for quick snapshots with a TC.  You need to
have time to carefully focus and use a remote release.  I find also that I
usually am using it for things that are further away rather than smaller and
the pictures lack contrast when I use a TC.  You can correct this somewhat
in photoshop, but you don't get the rich colours that you get without the
TC.  The increased chromatic aberration is easy to fix and a very small
issue.

I'd say put the money in savings for a longer lens.

I assume you want to use it on a DSLR. One problem here is the format
factor. Since the DSLR sensor is small than regular 35mm film, things
are magnified by the sensor. So DSLRs are already rather demanding when
it comes to lense resolution.

By adding a teleconverter you further enlarge any fuzziness or lense
errors. With a zoom, the result is most likely disappointing. While
teleconverters can be useful, that is usally only the case with really
expensive prime telephoto lenses.

Another problem is light. What is your aperture at 300mm? I guess it
could be f5.6, and with a 1.4x teleconverter that turns into a 420mm
f8, which is not all that useful unless you use a tripod or you have
really brilliant sun shine. AF probably does not work, and even the
focus indicator may give up.

Thomas