Solid navy and burgundy are two highly versatile ties for the items you mentioned.
This is an excellent observation. You might not want to wear a solid navy tie with a navy blazer, but what doomx said is generally correct.
Make sure they are not very shiny, so stick to repp (twill) or grenadines.
Right. Definitely stay away from too shiny/satiny, since ties with that characteristic do not go well with the tweed jacket and other casual clothes you mentioned.
Because you are looking for ties that work equally well with tweed sport coats and suits, you should start out with rep stripes. Classic, reliable, highly versatile rep stripes. They go with everything. Accordingly, I recommend you visit the men's necktie section of the Brooks Brothers website and, using the filter, look at just the "striped" ties. Study every striped tie you see so that you can get an idea of what a versatile tie looks like. Let the experience serve as your education.
I want you to pay special attention to the following ties, because they will perfectly serve your needs:
BB #1 rep
Mini BB#1 rep
BB #3 bar stripe
Mixed weave sidewheeler stripe tie
Natte double stripe tie
Argyle and Sutherland
Let me repeat that last one: Argyle and Sutherland. It's timeless and goes with everything. Argyle and Sutherland. No man's tie collection should be without it.
If the prices of the Brooks Brothers ties are beyond your budget, keep in mind that BB has frequent sales. Alternatively, you can look at other web sites for less expensive rep ties. They won't be identical to the ones sold by BB, except perhaps for the Argyle and Sutherland, but you should be able to find ones that are close in spirit to the BB ties.
And as Dr. Livingston stated, a solid burgundy grenadine is also extremely versatile.