In my book I´ve come to the subject of (adlaştırma) nominalization, which are the suffixes (eylem + -mak = ad) (eylem + -ma = ad) 

Examples: onu görmek beni Mutlu ediyor. 

                    Sabahları yürümek bana çok iyi geliyor. 

                    Burada Sigara içmek sakınca var. 

In these sentences, I´m having a problem understanding when to use, ma, mak/mek,  and meye suffixes. For example in the first sentence why not (onu görmeye)? In the second sentence why not (Sabahları yürümeye) and in the third why not (içmeye).? If these suffixes all give a noun meaning. What are the differences? 

Thank you. I´d really appreciate some explanations. 

Infinitives are verbal nouns. They are nouns and they behave like nouns.

Because they are nouns they take case endings like nouns.

               gitme|nin GEN

               gitme|yi ACC

               gitme|ye DAT etc.

To choose the correct form you must analyse the syntactic position of the verbal noun in the sentence. In your examples

the verbal nouns are subjects of the sentences. The case of the subject in Turkish is nominative/absolutive.

Examples with datives:

Seni görmeye geldim. (DIRECTION INTO > DAT)

Bebek yürümeye başladı. (BAŞLAMAK + DAT)

Sigara içmeye devam ediyorlar. (DEVAM ETMEK + DAT)

But, assuming the 3 example sentences are correct, why was -mek choosen rather than -meye?

Now that you mention it, I suspect there is a mistake in the third sentence. I think it should be

                             Burada sigara içmek sakıncalı.

Anyway, the reason is because they are subjects. Dative can never mark the subject. Compare with other types of subjects, no difference:

Onu görmek/bayram/yolculuk/o...beni mutlu ediyor.

Sabahları yürümek/yeşil renk/yalnızlık...bana çok iyi geliyor.

Burada sigara içmek/başörtüsü/olumsuz beklentiler...sakıncalı.

Thank you, makes a lot more sense to me now.