Categorical Regression

My apologies if I missuse any terminalogy.

I have an ordinal variable (4 possible values) which I want to predict
based upon approximately 20 sampled ordinal variables (2 - 7 possible
values). In particular, I hope to answer the following two questions:
1) Can I build a model that correctly classifies all of my samples
into these 4 categories based upon the samples
2) Can I obtain a measure of importance associated with the sample
variables

My understanding is that this is a categorical regression problem. I
use R periodically for 'regular' linear regression and I have searched
the help files and see no associated categorical regression functions.
Does R have a function for this procedure? If not, any advice on how
to approach this problem otherwise?

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