-le / -tzaile / -lari

From verbs: -le and -tzaile. These attach to the verb radical, the short form of the verb:

• -le usually follows a radical ending in s, z, tz, ts or r, and verbs ending in -n: ikasi → ikasle (student), irakatsi → irakasle (teacher), idatzi → idazle (writer), irakurri → irakurle (reader), entzun → entzule (listener), egin → egile (maker, author).
• -tzaile goes with most other verbs, especially those ending in -tu: saldu → saltzaile (seller), lagundu → laguntzaile (helper), antolatu → antolatzaile (organizer), erabili → erabiltzaile (user), ikertu → ikertzaile (researcher).

This is a strong tendency, not an exact rule, so learn the common words as vocabulary.

From nouns: -ari and -lari. These name someone whose job or hobby is the activity:

• -ari is usual after nouns ending in -a: denda → dendari (shop assistant), kazeta → kazetari (journalist), pilota → pilotari (pelota player), dantza → dantzari (dancer).
• -lari is usual after consonants and -i: kirol → kirolari (athlete), argazki → argazkilari (photographer), and also abeslari (singer), bertsolari (verse improviser), txirrindulari (cyclist).

Euskaltzaindia's rule 49 chooses -ari for longer nouns in -a: politikari, gramatikari. Nouns in -ia take -lari: historialari (historian), zientzialari (scientist).

Using them. These are ordinary nouns. They take the article and case endings: Nire ama irakaslea da. Saltzaileak prezioa jaitsi du. Basque has no grammatical gender, so the same word is used for men and women (see no grammatical gender). Other doer endings come later: -gile and -gintza and -zale and -kide.