First Declension
(1) Feminine nouns in -η

πύλη, a gate

(2) Feminine nouns in -α

Note 1—When a noun has the stem-ending - α preceded by a vowel or ρ, the singular retains α throughout, as follows:—

βασιλεία, a kingdom

Note 2—When a noun has the stem-ending -α preceded by a consonant, the α becomes η in the genitive and dative singular (α being kept in the other cases), as follows:—

γλω̂σσα, a tongue

(3) Masculine nouns in -ης and -ας

Note 1—Masculine nouns of the first declension in -ης form the genitive singular in -ου and the vocative in -α. In the other cases they are declined just like πύλη.

Note 2—Masculine nouns of the first declension in -ας also form the genitive singular in -ου and the vocative in -α. In other cases they are like βασιλεία (see above).

Note 3—The plural is the same throughout in all first declension nouns.

μαθητής, a disciple

νεανίας, a young man

The Second Declension—Stem ending -ο

Masculine nouns, and a few feminine, in this declension end in -ος. Neuter nouns end in -ον.

λόγος, a word

ἔργον, a work

Note 1—All neuter nouns have the same form in the nominative, vocative, and accusative.

Note 2—Observe the iota subscript under the dative singular.

Note 3—The masculine article ὁ must go with masculine nouns when “the” comes before, and the neuter τό with neuter nouns.

The Third Declension

Introductory Note—Nouns in this declension are of all three genders. There is a considerable variety and hence a number of paradigms are necessary, but all follow a simple form which presents little or no difficulty.
The essential thing is to know the stem, i.e., the elementary part of the word apart from the endings, or inflections. The stem can always be found from the genitive singular by taking away the inflection ending. Note that the genitive singular in the third declension usually ends in -ος. Take away the -ος and you have the stem. The stem will be a guide to the nominative case.

αἰών, an age (masc.)
stem, αἰών-

ῥη̂μα, a word (neut.)
stem, ῥηματ-

Notes
(1) The nominative and vocative are alike, and in the neuter the accusative also, as in the first and second declensions.
(2) The accusative singular ending -α was originally -ν, as in the other declensions, and the -ν is retained in several third declension nouns, the stems of which end in a vowel. These will be illustrated later. The -α ending should, however, be regarded as normal.
(3) The genitive singular ending is -ος, added to the stem.
(4) The dative singular ending is -ι, added to the stem.
(5) The nominative plural, in masculine and feminine nouns, ends in -ες, added to the stem. Neuter plurals always end in -α in the nominative, vocative, and accusative.
(6) The accusative plural masculine ends in -ας.
(7) The genitive plural ends in -ων, added to the stem. All genitive plurals end -ων.
(8) The dative plural ends in -σι, added to the stem, with various modifications. The -ν in brackets in the dative plural does not belong to the word; it is added at the close of a sentence, or when the next word begins with a vowel; this is simply for the sake of the sound.

Rule 1—The usual ending of the nominative singular is ς, added to the stem. The nominative endings provide a considerable variety and present a difference in form from that of the stem seen in the other cases. There are certain principles which govern the formation of the nominative, but these need not be learned. They simply serve to show that the variety of the third declension nouns is based on one form of case ending. The student should become familiar with the actual examples given and should keep in memory the other case endings, namely, -α, -ος, -ι, of the singular, and -ας, -ων, -σι, of the plural, as already learned in the noun αἰών.
κη̂ρυξ, a herald (masc.)
stem, κήρυκ-

The question arises as to why the nominative, vocative singular, and dative plural have an ξ, whereas the rest of the cases have a κ. The explanation is as follows:—
When the stem (here κήρυκ-) ends in κ, or γ, or χ (which letters are called gutturals), the addition of the ς to the stem produces the letter ξ in the nominative and vocative singular and the dative plural. Thus κη̂ρυκ with ς, makes not κη̂ρυκς but κη̂ρυξ. The other cases retain the κ-.

Ἄραψ, an Arab (masc.)
stem, Ἄραβ-

The stem is seen to be Ἄραβ-. When a stem ends in π, or β, or φ (which letters are called labials) the addition of the ς to the stem produces the letter ψ. Thus Ἄραβ- with ς makes, not Ἄραβς, but Ἄραψ.

παι̂ς, a child (masc. or fem.)
stem, παιδ-

We observe that the stem is παιδ-. Now whenever a stem ends in τ, or δ, or θ (which letters are called dentals) the addition of ς causes the dropping of the τ, δ, or θ. Hence παιδς becomes παι̂ς and παιδσι becomes παισί. Similarly ἐλπιδ- is the stem of ἐλπίς, hope. The student should write out ἐλπίς in all its cases from memory, on the model of παι̂ς.

Note—Nouns in the third declension whose nominative ends in -ις, -υς, -αυς, and -ους  usually have, in the accusative singular, a shortened form, ending in -ν. Thus while the stem of χάρις, grace (or thanks) is χαριτ-, and hence the genitive is χάριτος, and the dative χάριτι, the accusative is χάριν, but χάριτα exceptionally in Acts 24:27.

ἰχθύς, a fish (masc.)
stem, ἰχθυ-

This stem ends, then, in a vowel. When a stem ends in a vowel the nominative is formed by simply adding the ς; “a fish” is ἰχθύς.
Bearing in mind the note just given, that the accusative of nouns ending in -υς etc. end, not in -α, but in -ν, we have the following paradigm for ἰχθύς:—

Rule 2— (not to be committed to memory)—When a stem ends in -ν, or -ντ, or -ς the nominative is formed by lengthening the preceding vowel. The same is usually the case with a stem ending in -ρ.
Take for example, ποιμεν-, the stem of the word for “a shepherd.” The nominative is ποιμήν (note the η instead of ε); the accusative is ποιμένα, the genitive ποιμένος, etc. Note that the dative plural is ποιμέσι (not ποιμένσι—the ν was dropped before ς); again, λέων a lion (stem λεοντ-) has accusative λέοντα, genitive λέοντος, etc. The dative plural is λέουσι not λέοντσι—a combination too awkward for Greek ears; note the ω in the nominative instead of ο. So again with ῥήτωρ, an orator (stem ῥήτορ-), it has accus. ῥήτορα, etc. The dative plural is ῥήτορσι.
Note—One or two nouns ending in ρ are a little irregular. The two following must be memorized:—

πατήρ, a father

* (Note short ε)

Note—μήτηρ, a mother, and θυγάτηρ, a daughter, are declined in the same way. Write them out in full from memory, not forgetting the shortened form in the gen. and dat. sing., and the gen. plur., and the α in the dat. plural.

ἀνήρ, a man

Note—ἀστη̂ρ, a star, keeps the ε throughout (e.g., gen. ἀστέρος), except that the dative plural is ἀστράσι.

Rule 3—Some nouns ending in -ις and -ευς have a genitive ending with -εως instead of -ος. The two following should be memorized:—

πόλις, a city (fem.)
stem, πολι-

(Note the accusative in -ιν)

Like πόλις are δύναμις, power; κρίσις, judgment; ὄφις, a serpent, and others.

βασιλεύς, a king

Note (1) the ordinary accusative ending -εα, (2) the nom., voc. and acc. plural in -εες and
-εες; contract these double vowels to ει (for the sake of sound).
Like βασιλεύς are γραμματεύς, a scribe; γονεύς, a parent.

Neuter nouns of the third declension are important, and are of two chief kinds. Remember that all neuters have the same form for the nominative, vocative and accusative cases.
(1) Most conform to the example ῥη̂μα presented above.

Learn the following:—

There are a few words not ending in -μα which are neuter and come here, such as πυ̂ρ, fire (genitive πυρός); φω̂ς, light (genitive: φωτός); τέρας, a wonder (genitive τέρατος).
(2) Other neuters ending in -ος have some contracted endings. The following model must be memorized:—

γένος, a race, generation

Note 1—The genitive singular γένεος contracts to γένους; the nominative, vocative and accusative γένεα contract to γένη; the genitive plural γενέων to γενω̂ν.

Note 2—These neuters in -ος, must be distinguished from second declension masculine nouns ending in -ος like λόγος.
The student will soon become accustomed to the two varieties as found in the New Testament.

Note 3—These neuter plurals in -η (for -εα) must be distinguished from first declension feminine ending in -η, like πύλη. The context generally helps to distinguish.

Reference: Vine, W. 1997. Vine's you can learn New Testament Greek! : Course of self-help for the layman. Nashville: Thomas Nelson.