I have been thinking about how to approach this. I surmise that the largest problem is the flexibility of the meter: for each foot, you will have to decide wheather it is a dactyl or a spondee; this is (with very few exceptions) not a matter of dispute, but it is only natural that it takes some practice to learn to to recognize which it is. Of course, knowing all the vowel quantities by heart greatly helps in this regard; in fact, with a proper pronunciation, you could read any poetry and the meter would realise itself automatically. (Such a pronunciation is however very seldom heard, I might add.)
Until you have achieved fluency, you will probably have to analyze each verse methodically. As you already have noticed, it is often helpful to start from the end, due to the regularity there. It is, however, of course very unnatural to read backwards, so I would advise you to try to attack the verse from the left, to begin with, and save the begin-from-the-end approach to the otherwise hopeless verses.
Let’s see how one may do it in a concrete example. (And you have to forgive me if I’m too thorough in what follows.) If I understand your syllabus correctly, you may have chosen to read, for example, Ovid’s rendition of Orpheus and Eurydice? We can take that text in any case:
(…nupta per herbas)
dum nova Naïadum turba comitata vagatur
All hexameters start with a long syllable, in this case dum (not because the u is long, for it isn’t, but because the next word starts with a consonant). Then two things can happen: either the foot is a dactyl, and two short follow, or it is a spondee, which ends in one long. Which is it? In other words, is no a short or long syllable? If you don’t know, try looking at the following syllable, va. If it is long, then no must be long as well. If it is short, then no must be short as well. (Why is this?) In this case this doesn’t help, though, since you can’t know (at first) whether nova is nominative or ablative.
This is where the proper pronunciation come into play, which I was speaking about above: chance is that you have heard this word pronounced nōva, which is not correct: the classical pronunciation was with a short o, which you will find if you look it up. Knowing this, we understand that va is a short syllable as well.
Next foot: the trema over ï means it is to be pronounced as a separate vowel (and not as part of a diphthong); i.e. the syllabification would be Na-i-a-dum. The first syllable must be long: Nā. Then we have the same problem again. (Come to think of it, this verse was a fairly bad example to begin with, but when I have started with it, I might as well continue.) The Swedish pronunciation may lead you astray, but if you look it up (under Naias), you will find that it is Nāĭădum.
So far, this might not have been so helpful, but the next foot is a better example: it starts with -dum, which is long, again because the next word starts with a consonant. Then comes tur-ba. Same thing here: either this is two long, or two short syllables; thing is, you see immediately that tur is a long syllable (by position)! Syllables long by position are your friends. 
Hence bā must be long as well, and starts foot number four. (We have an ablative form here, evidently.) Then co-mi; two long or two short? You can look it up, or you can look at the end of the verse, and hopefully see that the two last feet are -tā-tă vă-gā-tur, which means that it must be cŏ-mĭ.
You can solve all this without having to look things up, by trial and error. For example, you might attempt something like this (* signifies it is incorrect; accents denote the ictus, the first syllable in each foot):
*** dúm nōvá Naiádum túrbā cómītáta…** but then you run out of feet before you are done. Some of all these spondees must be dactyles! In that way, we will be able to “consume” more syllables (since a dactyl is one more syllable than a spondee). We try again:
*** dúm nōvá Naiádum túrba comítātá va…** No, still too many syllables. Can it really be comităta? Let’s try:
*** dúm nōvá Naiádum túrba comítata vága…** Darn. How can we improve this? In this version we have two spondees (dum no- and -adum), but we can’t do anything about the later, because that last syllable is long by position. It must be that the verse actually starts with a dactyl! We were wrong all along. Ah well, then we have solved it, no?
*** dúm nŏva Náīádum túrba comítata vága…** Still we have two spondees, which is one too many, but we can only change the first:
dúm nŏva Náiadúm turbá comitáta vagátur. There!
This was fun, let’s take another one! 
occidit in talum serpentis dente recepto.
Compared to the first one, this is a walk in the park.
Does the word oc-ci-dit end with two short or two long syllables? Or, let’s put it this way, how would you accent it (in prose)? (Where the accent is placed depends on the syllable lengths, as you know.) Not sure? Well, look at the final syllable of the word. It is not long by position, since the vowel is followed by only one consonant in total, so if this syllable is long, then the vowel must be long. But, if you know your endings, then you know that it should be short. Hence also óccĭdĭt.
Next foot starts with in, long by position. Then ta-lum; of these the last is long by position, since the following word starts with a consonant; hence tā is also long, and ends this foot, which is a spondee.
The third foot is -lum ser-, two syllables, both long by position. Spondee.
Fourth foot: -pen-tis. Again, two syllables long by position. Spondee.
And then you know the rest: den-tĕ rĕ-cep-to.
This verse is so full of syllables long by position, that the only place where uncertainty can occur is in the first word. What happens if you read it wrong?
*** óccidít in tálum sérpentís denté re…** Not only do you run out of feet to contain your syllables in, you are also forced to lengthen the -e of a third declension ablative, which, if you know your endings, should be short.
We can take one more:
Quam satis ad superas postquam Rhodopeïus auras
You know the drill: you will have to decide whether the second foot begins on -is or on ad. The later is long by position, but the first would be long only if the vowel were long, i.e. if the word were pronounced sātīs. Is it? Unfortunately, just as with nova, Swedes are prone to lengthen stressed syllables, and more often than not pronounce this word as sātĭs, which might lead you in the wrong direction… Looking it up, you will see it is sătĭs, and knowing this, you should be able to solve the rest.
Let me finish with talking about about stress. Correct me if I’m wrong, but you will probably have been taught to scan (“skandera”) your poetry, i.e. stress the ictus? Since Swedish phonology automatically implies that stressed syllables are long and unstressed syllables short, this has the effect that the correct rhythm is automatically achieved, at the expense of the natural word stress. I’m not an enemy of this practice as such, but you should be aware that this is merely a crutch, and that Latin poetry originally was read both with the correct rhythm and with the word accents in their normal places. Hopefully your professor has stressed that last part (no pun intended).
To achieve this, you will have to learn to pronounce sátis with a short stressed ă, and at the same time refrain from lengthening the t. For, as I said, Swedes are prone to lengthen stressed syllables, and if they don’t say sātis they will be inclined to say săttis, which is just as wrong. It should be să-tis with two equally short syllables. This is not really extremely difficult – you probably pronounce stressed short syllables everytime you speak English, unless you have a rather heavy Swedish accent – but, from my experience, this is seldom taught in Latin.
In the same way, you will have to learn to pronounce long unaccented syllables (“turbā”). Might be a bit more difficult? I don’t know.
But, all this I have said about stress is probably above what is expected of you. I’d advice you to learn to scan comfortably first, and then later, if you are so inclined, try to learn to combine normal word stresses with the poetic rhythm unaltered.
P.S. If you actually do read Orpheus and Eurydice, here is a tip for you: in the line starting with “Persephonen”, you must read “ădĭīt”, with a long final ī. This is an arcaism sometimes employed by the poets.