As a native, "toward" is pronounced exactly like "to ward", but (usually) with the highly-unstressed vowel variant of "to". Remember that "w" is a semivowel, but it's not doing anything special here (at least in the vast majority of mainstream English dialects). In contexts where it is emphasized (or I suppose in more formal registers) it can strengthened to merely the normal lack of stress.
English might make more sense if someone actually sat down and wrote out the real stress rules, rather than trying to cram everything into just "unstressed" and "stressed" and only caring within a word.
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"To" might be one of the syllables with the most possible stress levels, with at least 4 and possible more. As I spell them,
1. "too" - full stress. Common for "two" and "too", but possible for "to" under rare circumstances.
2. "to" - less emphasized but still arguably stressed; still has the "proper" vowel. Usually this is as strong as "to" gets; "two" and "too" often fall down to this level if before a stressed syllable. Arguably this could be split into "stressed but near words with even more stress" and "unstressed but still enunciated" (which occurs even within a register).
3. "tah/tuh" - unstressed, the vowel mutates toward the schwa. Very common for "to", but forbidden in a few contexts. May be slightly merged into the previous syllable. Can we split this?
4. "t'" - very unstressed vowel has basically disappeared; may or may not remain a separate syllable from the one that follows (should that be split?).
The infinitive particle can't be 3 (normally 2, not sure if 1) if the following verb is implied (but not if the speech is cut off). At the start of the sentence it also can't be 3, and 1 is possible as seen below though 2 remains the default. Note that many common verbs act specially when before an infinitive particle; although sometimes treated as phrasal verbs it would be silly to treat them as taking a bare infinitive as their argument.
Adverbial particle "to" when the phrasal verb takes a direct object can be 2 or 3; this likely depends on the specific verb it's part of. Note that many people parse this as a preposition (taking a prepositional object), but this is technically incorrect (though there are some verbs where it really is unclear even when doing the rearrangement and translation/synonym tests).
Adverbial particle "to" when the phrasal verb does not have a direct object is usually 2 or even 1 (e.g. in the imperative). Some heretics have started calling this a preposition too (unfortunately, often in ESL contexts), but this should be avoided at all costs; they're just too cowardly to give particles the respect they deserve. Probably the only common example in modern English is "come to", but there are several others in jargon or archaic English.
Particle/preposition (the parsing is arguable) "to" used between numbers (range, ratio, exponentiation, time before the hour) tends to be 3, especially if one of the numbers is a "two". With variables it is slightly more likely to be 2.
Preposition "to" meaning "direction", or "contact", or "comparison/containment" tends to be 2, but can usually fall to 3 (less likely at the start of a sentence, and can also be prevented by what precedes it, e.g. "look to" can fall to 3 without much effort, but "looked to" strongly stays at 2). Contrast with "toward" of related meaning, which takes effort to get from 4 to 3.
Preposition "to" meaning "according to", "degree", or "target" (including but not limited to the explicit expression of an indirect object with most verbs, which we could argue should count as a particle instead. If you're wondering what verbs are excepted, one is "ask" - it can only use "of", as in "ask a question of him") is much more strongly 2, and requires significant effort to force it down to 3.
Adverb "to" is always 2 I think, but this is rare enough that I'm not sure.
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"To be or not to be", as famous as it is, has a pretty unusual stress pattern for most of its words: full stress on the first "to", semi-stress on the first "be", no stress (but still full length) on "or" (normal), full stress on "not", some stress on the second "to", and some stress on the second "be" (more than "to" but less than "not").