You have a multi-class classification problem with 1728 samples. The features are in 6 groups:
buying v-high, high, med, low
maint v-high, high, med, low
doors 2, 3, 4, 5-more
persons 2, 4, more
lug_boot small, med, big
safety low, med, high
what you need to do for features is to create features like this:
buying_v-high, buying-high, buying-med, buying-low, maint-v-high, ...
at the end you'll have
4+4+4+3+3+3 = 21
features. The output classes are:
class N N[%]
-----------------------------
unacc 1210 (70.023 %)
acc 384 (22.222 %)
good 69 ( 3.993 %)
v-good 65 ( 3.762 %)
You need to try several classification algorithms to see which one works better. For evaluation you can use cross-validation or you can put away say 728 or the samples and evaluate on that.
For classification models you iterate over 10 different classification models available in Machine Learning libraries and check which one is better. I suggest using scikit-learn for simplicity.
You can find a simple iterator over several classifiers in this script.
Remember that you need to tune some parameters for each model and you shouldn't tune them on the test set. So it is better to divide your samples into 1000 (training set), 350 (development set), 378 (test set). Use the development set to tune your parameters and to choose the best performing model and then use the test set to evaluate that model over unseen data.