Greek I; lesson 14

John 1:29

Τῇ ἐπαύριον βλέπει τὸν Ἰησοῦν ἐρχόμενον πρὸς αὐτὸν

επαυριον is an adverb. Which of the adverbial questions does it answerwhen?? What verb is modifiedβλεπει? What is the DO of βλεπει?Ἰησοῦν  
ερχομενον
ερχομενον is anarthrous. Hence, it can be either adjectival or adverbial. Here, ερχομενον is attributive. It is modifying Ιησουν. Think of it this way, Jesus is the “coming to him” Jesus. When you put it like this, you can see the attributive nature of the participle more clearly.
Attributive participles are best translated into English using an adjectival dependent clause. In this case, “The next day, he sees Jesus who was coming to him.”

 

καὶ λέγει· ἴδε ὁ ἀμνὸς τοῦ θεοῦ ὁ αἴρων τὴν ἁμαρτίαν τοῦ κόσμου.

Ιδε here is an interjection. It has no other connection to any other part of the sentence. This also explains why αμνος is is in the nominative case. You need not spend any time on this; but if you wish, you can see an explanation in GGBB p. 60.
θεου is probably an ablatival genitive. Use GGBB p. 72 to decide which category best fits the context here.The choices are Genitive of Separation, Genitive of Source (or Origin), Genitive of Comparison  
αιρων
Check the participle memory forms to parse αιρων. αιρων is another attributive participle. Again, think of it as this Lamb is the “removing the sin of the world” Lamb (αμνος). Attributive participles are usually translated into English using an adjectival dependent clause. In this case, “The Lamb of God, Who removes…”
 

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John 1:30

οὗτός ἐστιν ὑπὲρ οὗ ἐγὼ εἶπον·

In your translation, you will need to provide a PN for the verb εστιν. “This is [the One] on behalf of whom…”
ου εγω ειπον is a dependent clause. Here, however, it is not adjectival or adverbial but a noun clause. It is the OP of υπερ.
Lexical form of ειπον is λεγω.
 

ὀπίσω μου ἔρχεται ἀνὴρ ὃς ἔμπροσθέν μου γέγονεν,

οπισω is an adverb here used as a preposition.
What is the subject of ερχεταιἀνὴρ? What is the dependent clauseὃς ἔμπροσθέν μου γέγονεν? Which noun does the relative clause modifyἀνὴρ?
 

ὅτι πρῶτός μου ἦν.

How did you translate οτιhopefully with the DMW 'because'? Is this clause adverbial or adjectivaladverbial answering the why? question?
πρωτος is a superlative adjective here used as a comparative (GGBB p. 303).
 

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John 1:31

κἀγὼ οὐκ ᾔδειν αὐτόν,

καγω is an interesting conglomeration of και and εγω.
ηδειν is a pluperfect form; first person, singular.
 

ἀλλ᾽ ἵνα φανερωθῇ τῷ Ἰσραὴλ διὰ τοῦτο ἦλθον ἐγὼ ἐν ὕδατι βαπτίζων.

ινα φανερωθη τω Ισραηλ is a dependent clause. The DMW indicates an adverbial function. The clause is answering the why? question. What mood is φανερωθη (principle 27)We always expect the subjunctive mood after the conjunction ινα.?
In your translation, you will probably put διὰ τοῦτο ἦλθον ἐγὼ ἐν ὕδατι βαπτίζων before ἵνα φανερωθῇ τῷ Ἰσραὴλ.
βαπτιζων is anarthrous. This leads us to think it might be an adverbial participle. Recall that one of the adverbial questions is “how?”. Now ask yourself, how did he come? He came baptizing.
How do you know that δια means “on account of” and not “through”because the OP is in the accusative case?
φανερωθη

φανερο-      stem
φανεροθη       tense formative
φανερωθη        principle 12
φανερωθη          verb ending (there is none for 3rd sing.)