What did the early Church Fathers (pre-Nicea A.D. 325) have to say about faith, baptism and if it is the act of baptism that saves?

They said an awful lot - far too much to detail here! One way of finding an answer is to read explanations from official Catholic sources as to developing changes regarding the connection between faith and baptism with regard to salvation, as to whether one or the other saves.

Some Christians believe that BOTH are needed, others believe that as long as you have faith in Christ you don't have to be baptised to be saved, while yet others say that baptism is the key to salvation as you can't be saved without it. But as you ask about about early Church teaching, pre-A.D. 325, that narrows the quest down considerably. For a start, here are a few points from a Catholic article explaining how a "developed", or "mature" view on baptism arose:

"The baptism of adults is presupposed throughout the NT and the early Christian period... Infant baptism developed naturally out of the entirely different circumstances in which Christianity found itself when society had become Christian... But child baptism became the normal practice from about the 4th or 5th century onwards... even today, through the intermediary of their godparents infants are treated at their baptism as if they were adult: they [the godparents] renounce Satan, confess the faith, and state that they wish to be baptized." Encyclopedia of Theology, p69, article by Burkhard Neunheuser, Burns & Oates, 1961

Of interest, Tertullian condemned an instance of infant baptism in 197. Despite that, baptising infants gained ground. However, the quote above shows that baptism of babies and infants was not encouraged by the earliest Church Fathers (though it did happen). Around 200 a baptismal liturgy was written down, and its development can be traced via Justin's Apology (I, 61), Tertullian's treatise on baptism, and especially Hippolytus's Apostolic Tradition (circa 200). Clearly, babies and infants cannot do what is required of these baptismal candidates. Quoting again from the previous source:

"First comes a lengthy catechetical preparation of the candidates; next an immediate preparation of fasting, prayer, and solemn promises, then the solemnity of baptism itself. Baptism proper is an actual bath in flowing water. The candidate is thrice immersed, and each time one of the divine names is pronounced (epiclesis). Finally comes the anointings and the imposition of hands. The new Christian is now admitted to the common worship of the faithful, to the kiss of peace, and to the Eucharist. These basic lines were kept unchanged subsequently. The ritual added a confession of faith, the renunciation of Satan and the baptismal promises." (Ibid. p. 68, bold italics mine)

This article suggests that a person only became a Christian at baptism, and so only then able to be admitted to worship, but I am not able to say if that is so. If a confession of faith at baptism was only added later to the liturgy for baptism, the question arises as to whether faith was viewed as the first step that would lead to baptism (as an evidence of having faith). A problem here is that the Catholic article I'm using is not clear (to me). When I turned to read its 57 columns of text on 'Faith' (pp.496-510) I finally hoped to find an answer under the sub-heading, Faith and salvation (p.508). Here are some points:

"The role of faith in salvation is fundamental (D 801). By faith man recognizes the reality and the absolute gratuitousness of God's initiative in saving sinful humanity through Christ (Rom 10:9; 3:22-30; 4:16; Gal 2:16; 8:22, 24; Eph 2: 8-10). Man can only be saved by sharing in the mystery of Christ's death and resurrection, which means that he must first of all affirm the reality of this mystery (1 Cor 15:12-16) and freely accept the economy of salvation that God has established and revealed in his Son (Rom 10:16; 2 Cor 9:13; 2 Thess 1:8). Thus faith is man's response to God revealing himself as man's saviour. It is an attitude which accepts salvation as sheer grace, renouncing all pride in one's own works (Rom 3:22, 24, 27; 4:2, 20: 1 Cor 1:29; 4:7; Gal 2:16). Man's response to the absolute gratuitousness of God's saving intervention is faith. Hence faith confesses the grace of God." (Ibid. p. 508, article by Juan Alfero)

Now, that article was published in 1981, not 198! The reason for its inclusion is that the early Church Fathers (who went by the scriptures) would likely agree with all of the above re. the Faith and salvation sub-heading, and understand why those many Bible texts supported such a stance on faith in relation to salvation. However, I found nothing in that 1981 section that mentioned baptism, let alone connected it with salvation.

The "evolution of the baptismal liturgy" into "the form we now have in the Roman Ritual" (Ibid. p. 68) includes views such as descent into the waters washing away one's old morality (i.e. sin), ascent from the waters being the passage from death to life. For that point an ancient source is given, but not a biblical one. It is Pseudo-Barnabas; Shepherd of Hermas.

"Thus baptism is a bath that washes away sin, the free remission of the penalties of sin and illumination on the say of salvation. It perfects and seals us, transports us over the frontier of death into the life of Christ (St. Clement)". (Ibid. p. 69)

Origen fits these truths into his own profound view of the history of salvation... the outward form of baptism derives its meaning from the spiritual realities... gives us the grace of Christ, and thus bears us onward to the final stage of baptism, the final resurrection from the dead... It completes sacramentally the ascetical death of the catechumen; but 'If anyone comes to the washing of water (continuing) in sin, then his sins are not remitted' (21st homily on Luke). (Ibid. pp.69-70) [Note, Eastern Orthodoxy views Origen as a heretic due his views on dualism]

Given that those articles give the modern Catholic view of developments from the early Church till now, the desire to show continuity will be present, so if there is discontinuity, another source will be needed to show this. Hopefully, other answers will provide more precise quotes from the early Church Fathers.