Teanglann has recordings for a lot of words. You could have a listen to the difference between bó [bˠoː] ("boh" cow) and beo [bʲoː] ("byoh" alive), or cad [kadˠ] ("kad" what) and cead [cadˠ] ("kyad", permission). You can also check Wikipedia, which has information about how various sounds are pronounced.
Very basically, you can think of slender consonants as having a slight hint of "y" after the consonant. This can get tricky at the end of a word, but you can get around it by keeping your mouth open at the end, instead of closing it. For broad consonants, leave your mouth hanging open like for "uh"; for slender, leave it in the "i" position. Kind of like ending with a vowel, but without pronouncing that vowel: leabhar [lʲəuɾʲ] "lyowrᵘʰ", vs. leabhair* [lʲəuɾʲ] "lyowrᶤ".
Consonants are always slender before "e" and "i" in English (which is why English speakers don't notice themselves doing it), so to make them broad, you insert a "glide" in front (like a "w", but without rounded lips for consonants that aren't pronounced with the lips). So bí [bʲiː] ("bee", be) vs. buí [bˠwiː] ("bwee"), yellow. And saor [sˠɰiːɾ] ("sweer", free).
Annoyingly, there don't seem to be many sources for palatalization (making sounds slender) in Irish, but Russian has the same feature, and it's quite well explained in a lot of places: http://therusblog.com/2012/02/18/palatalization-in-russian/
Here are some diagrams for what to do with your tongue for palatalization (slender) and velarization (broad): https://www.quora.com/What-is-the-difference-between-hard-and-soft-consonants-in-Russian
Nualeargais is specific to Irish, and hopefully thorough without being too technical: http://nualeargais.ie/gnag/phonol.htm
The link doesn't go to the Lenition 'tips' page for me, just the whole skill tree, perhaps they've re-organised the site?
I did check out the Lenition tips though and it doesn't explain why there's no lenition of mairteoil following bhur. It does use mairteoil as an example of a feminine noun being lenited by the definite article, and of it causing lenition in a directly following adjective, (which also has to be feminine?).
In the tips "3. Possessive Adjectives" it states "Lenition occurs after mo, do, a"; but no exception is mentioned for this feminine noun following bhur. Is it because bhur is plural? The possessive adjectives cited are all singular I think?
Or is mairteoil just one of those exceptions that has to be remembered?
I did a bit more digging and did find the answer in Tips for this skill (Possessive) and in Eclipses.
Bhur causes eclipses not lenition, regardless of the noun being feminine, however words beginning with 'm' are not eclipsed.
I still suspect though that the division between whether something is lenited or eclipsed at all by possessive adjectives has something to do with singular and plural?
It really isn't any more complicated than that - The singular possessive adjectives lenite, the plural possessive adjectives eclipse. The gender of the possessed noun is irrelevant.
The obvious wrinkle is that the possessive adjective a can mean "his", "her" or "their". It lenites when a is the singular "his". It eclipses when a is the plural "their". But when a is the singular "her", there is no mutation of the following noun.
The singular possessive adjective mo and do become m' and d' before words that start with a vowel sound, - this isn't optional. The plural possessive adjectives ár, bhur and a cause an n-prefix for words that start with a vowel. a meaning "his" doesn't mutate words that start with a vowel, a meaning "her" causes a h-prefix for words that start with a vowel.
Because mo and do normally lenite, and fh is silent, mo and do become m' and d' before words that start with f+vowel - so m'fhoclóir but mo fhreagra.