For example, for the word bonus, bona, bonum ('good') the masculine nominative singular obviously has the same stem as the oblique forms. But, with a word like 'our' (noster, nostra, nostrum) how can you identify the stem with which to decline the word?
There are exceptions to the rule, but like nouns you can often predict what the stem is from the genitive declension of the adjective.
With your example, the genitive singular endings of noster, nostra, nostrum are
nostri -i (m)
nostrae -ae (f)
nostrum -i (n)
If you drop the -i or -ae ending, you'll get nostr as the stem. For third-declension adjectives, drop the -is ending.
For more information, see this question about how to find the stem of a noun.
An adjective almost always uses only one stem for its entire declension, excluding degrees. The genitive singular of any gender will tell you that stem. Just be careful about UNUS NAUTA adjective, or one of the naughty nine. They have genitives in -ius and datives in -i, but otherwise are normal.
Here are some examples:
The 1st/2nd declension adjective ruber, rubra, rubrum has genitives rubri, rubrae, rubri. The stem is very much rubr-.
the 3rd declension adjective vetus has genitive veteris for all genders. The stem is veter-.
Neuter, neutra, neutrum, an UNUS NAUTA adjective, has the genitive neutrius for all genders. Since we know that -ius is its genitive, we can say that the stem is neutr-.
Once you get used to the way adjectives tend to decline, you probably won't need to look at the genitive for every adjective.
I'd check out Allen and Greenough's Latin Grammar for more, like this section on 3rd declension adjectives. This section is more to give you an understanding of the possibilities, even if it doesn't always go into the reasons very much. I very much recommend it, since most people would not find it helpful to understand why the system is the way it is.