Then God said, "Let us make Man in our image, according to our likeness, so that he may rule over the fish of the sea and the birds of the sky and over the beasts and over the whole earth and over every creeping thing that creeps on the earth". So God created the man in His image; in the image of God He created him; male and female He created them.
Genesis makes it quite clear that this creation of Adam and Eve on the sixth day is the creation of human beings. Scripture never speaks of "a pre-Adamic race" – that is a construct of the imagination of certain interpreters who are trying to explain various biblical passages, especially Genesis chapter six (see below). Moreover, whatever preceded the re-creation of the earth during the seven day of Genesis chapter one will have to have taken place on the far side of the Genesis gap; there may be fossil evidence for creatures who, from their long-ago ossified bones, look to some "scientists" to be similar to human beings, but these would have been destroyed in the far distant past when the Lord judged Satan's rebellions and turned the lights out on the universe (before this there was no darkness) and subjected the earth to a deep-freeze that was only reversed during the period of re-creation. Consider the great apes; just because a creature has a skeleton similar to man does not mean it has a human spirit, free will, and the image of God. That cannot be deduced from bones or rocks.
I think it is a red-herring for people to demand something like "show me where in the Bible it proves there was no _____ ", as a necessary proof of non-existence (as some seem to be doing with you). As every practiced rhetorician knows full-well, it is impossible to prove a negative in such circumstances. The Bible never mentions many things which do not exist; the Bible never mentions aliens and still they do not exist: earth is the place God made for mankind (and originally for angel-kind as well); this is the theological center of the universe. Certainly, we do not need the Bible to specifically state that certain fanciful categories of imagined creatures do not exist to know that they do not exist (otherwise we be spending all our time trying in vain to find a scripture to prove that ghosts and goblins and elves and vampires and bigfoot etc. don't exist). It is of course is a bit easier in this case, since scripture is very clear about the actual origin of all mankind.
As to "giants", this term occurs in the KJV as the result of an unfortunate translation choice. The Greek version of the Hebrew Old Testament (i.e., the Septuagint or LXX), in order to make the Hebrew term nephilim (lit., "fallen ones") understandable to an audience versed in classical culture, translated the word as gigantes (finding a rough parallel to the nephilim in the Greek myth of the giants). KJV picked that up, but now people are mislead by the translation since 1) what we think of as a "giant" today is quite different from the Greek idea and 2) people today tend to see the KJV as almost inspired, little realizing that the translators expected an educated audience to understand that this was just a translation (one which was accommodating a difficult Hebrew term to a more familiar mythological one). In short, "giants" in the KJV are actually nephilim – these creatures were not necessarily larger than other people nor different in appearance (whereas the Greek giants were monstrously large and grotesque, having a hundred hands each!). It was their origin that made the nephilim so different: as half-human half-angels, there were exceptionally powerful and gifted individuals (although hostile to God and horrific from the divine point of view), "mighty men which were of old, men of renown" (giboriym asher meol'am anshey hashem).
You are correct in your assumption that these all perished in the great flood. Goliath and his ilk were abnormal in size but that is all (i.e., they were 100% human). Other Canaanite peoples are recorded as being likewise far above average in their physical size, but they too are only abnormal humans with no exceptional powers. The spies who entered Canaan and brought back the bad report to the children of Israel do call some of the peoples there nephilim, but they are clearly only using the word in a propaganda sense in order to dissuade Israel from entering the land. One of the reasons for the flood was that Noah and his family were apparently not only the last to fear God but also the last (or among the last) to still be of pure human blood (Noah is called "perfect in his generations" for a reason). Had God not acted when He did, the human race would have been effectively extinguished in that no pure human beings would have been left, and then the Messiah could not have been born (since Jesus had to be of pure human seed: cf. Gen.3:15: "her Seed").
I have written all this up elsewhere in detail and invite you have a look at the following links:
The Nephilim in Genesis 6
"The Nephilim" (in SR 5)
The Origin and Fate of the "Giants" of Genesis Chapter Six.
Antichrist and the Nephilim
Dinosaurs, the Nephilim, Noah, et al.
Eschatology Issues: The Nephilim
Doubts about the Nephilim
Thanks for your interest in this ministry. Incidentally, I maintain an email announcement list for major new postings to the site. If interested, I would be happy to add you to it.