The choice of model depends on what you are trying to accomplish. There are many estimation strategies available, including multinomial logit (MNL), multinomial probit (MNP), nested logit (NL, which is a specific instance of Generalized Extreme Value models, GEV), and mixed "flavors" of the aforementioned models. I think it is worth mentioning that despite advances in discrete choice models (a lot of this work was done in the later parts of the 20th century), MNL model remains a popular choice for modeling discrete outcomes for the reason that it is computationally simple. However, one key assumption of MNL, the independence of irrelevant alternatives (IIA), is often violated. In layman's terms, IIA assumes "equal competition" between outcome choices. Essentially, if A is preferable to B, then adding a third choice, C should not make B preferable to A (adding alternatives should not affect preference). This is often illustrated with the "red bus/blue bus" problem:
Suppose commuters have a choice between taking a car or a red bus to work. If these were the only two choices, then the probability of choosing mode of transport is equal, that is:
$P(car)=P(red bus)=0.5$, giving us an odds ratio of 1
Suppose that a third option--a blue bus--is now available. In real world situations, it is safe to assume that commuters probably don't care what color bus they take if they prefer to take buses. In this scenario, $P(car)=0.5$, and $P(red bus)=P(blue bus)=0.25$. The addition of the blue bus no longer makes the odds ratio of choosing car over red bus equal to 1, now that the probability of choosing the red bus has changed (probability now smaller with the introduction of the blue bus, which can be taken as a perfect substitute of the red bus).
By contrast, IIA assumes that the addition of the blue bus will not affect selection probabilities, that is, under IIA, it is expected that $P(car)=P(red bus)=P(blue bus)$, which in this case would equal to 0.33. This is not realistic, since people are likely not going to change their preferences (be less likely to take the car) because of the color of the bus.
The good news for you is, you can formally test whether the IIA assumption was met using the Hausman-McFadden test, and your statistical package likely has that test built in (See summary in Train 2009 [and Hausman and McFadden (1984) if you're interested in the gory details]). The other discrete choice models I described above (MNP, mixed MNL, NL) relax the IIA assumption, so those are viable alternatives for you. I also read a working paper a couple of years ago that compares the MNL with MNP and mixed MNL models, which concluded that MNL can still be used even if IIA was severely violated (see Kropko 2010).
I mentioned at the beginning of this response that choice depends on what you are trying to accomplish. If this analysis is for a class project, then you can probably get away with MNL and testing the IIA assumption as described above. If you are trying to publish this in a peer-reviewed journal, then you need to know your audience: the more purist the audience in their methods (statisticians, econometricians, etc.), the less likely they will be satisfied with MNL if the IIA assumption was violated. More applied journals are probably less concerned with the technical details (although I advocate doing it the "most correct" way, even if you are not going to be grilled too much about your methods).
You mentioned that you have price and quality data for each alternative. You can most certainly model those with MNL (if you choose MNL over other methods) by entering the price of the alternative as a separate covariate in the model (e.g. $price_A$, $price_B$, etc.) if your data are in wide format (one line in your dataset for each decision maker). OR, you can model using alternative-specific conditional logit (ASCL) if your data are in long format (each decision maker has the same number of lines as alternatives). MNL and ASCL basically give you the same results (Cameron and Trivedi 2009).
If you are using Stata, use
-mlogit- for multinomial logit and
-asclogit- for the alternative-specific conditional logit model.
Cameron, C and Trivedi P (2009). Microeconometrics using Stata.
College Station, TX: Stata Press.
Hausman, J and McFadden D. (1984). Specification tests for the
multinomial logit model. Econometrica, 52, 1219-1240
Kropko, J. (2010). A comparison of three discrete choice estimators.
Train, K. E. (2009). Discrete choice methods with simulation (2nd
ed.). New York, New York: Cambridge University Press. The complete
book is available on Train's website (which @gung linked to above).
Here's the direct link to the book chapters: