How intermarried couples are treated by the Jewish community:
Traditionally, each intermarriage is seen as a grave tragedy, because even if the children are technically Jewish, they are unlikely to have a strong Jewish identity and will probably intermarry themselves. A Jew who marries a Christian is likely to have grandchildren or great-grandchildren who are mainly Christian (or at least non-Jewish). This is very painful for many with a connection to Judaism, because Jews have maintained their faith through thousands of years of exile and persecutions (the worst of which were perpetrated by Christians).
However, the practical answer to your question depends on what Jewish community you are referring to. The Jewish community is made up of a number of different groups. For unaffiliated, unreligious Jews, intermarriage is pretty normal. So they would probably not treat you differently. If you tried to convert them to Christianity, this would be different, as even secular Jews would not take kindly to that.
For Reform or Reconstructionist Jews, intermarriage may be discouraged in general but in practice is common. Synagogues and other religious groups affiliated with these denominations have been trying to reach out to intermarried couples to encourage them to be involved in the Jewish community in some way (to attend services, send their children to Jewish schools (whether or not they are Jewish under Jewish law), attend community Jewish events, etc.
For this reason, these denominations probably treat intermarried couples as they would anyone else, even if the non-Jewish spouse has no intention of converting. (In these denomination, the conversion to Judaism of a non-Jew already married to a Jew is common, and is usually not a particularly difficult or lengthy process. However, such conversions would not be accepted by Orthodox Jews (or many Conservative Jews), because they are not done in accordance with Jewish law.)
Under Conservative Judaism, while intermarriage is forbidden, if an intermarried couple were to start attending a Conservative synagogue, they would probably be welcome (even if the non-Jew may not formally be able to become a member), because synagogues in general are in need of more members and do not want to discourage anyone. Conversion to Judaism of a non-Jew already married to a Jew is also fairly common in Conservative Judaism, though the process tends to take longer and has more requirements in terms of knowledge and observance (and would not be accepted by Orthodox Jews).
Orthodox Judaism forbids intermarriage, and it is rarer than in other denominations. While intermarriage is considered a sin, and a very consequential one due to the fact that the children will either be non-Jewish or will probably not identify as Jews, I'm not sure if that would mean Orthodox Jews would actually treat such couples poorly on a personal level. In practice, it is fairly common for a non-Jew already married to a Jew to convert Orthodox. Even so, this is a long process that requires becoming completely observant, and can cause marital difficulties if one spouse is less interested than the other.
Within Orthodoxy, Chabad is a chassidic movement focused on outreach to non-Orthodox Jews. Their goal is to encourage the fulfillment of mitzvos (commandments) by all Jews, and every mitzvah is considered precious. For this reason, they are very accepting of everyone and do not pressure anyone to become completely observant. It does often happen that an intermarried family (or the Jews in the family) will at least occasionally attend Chabad events or services (without fear of being mistreated or ostracized).
Effects on the children:
Will the children be raised as Christian, Jewish, both, or neither? Regardless, having parents of different religions can be a source of difficulty for the child, who may feel unsure about his or her identity, and may feel guilty for choosing one of the religions (as if it would mean rejecting the other parent). It also may lead to marital conflicts on the subject of how to raise the children, which can affect the children as well.
In terms of Jewish law, there is at least one effect having a non-Jewish father could have on the children. If you have a daughter, she would not be able to marry a kohen. (However, under the law if she marries one anyway they are not required to divorce). This would only be relevant if the daughter decides to become Orthodox (as the other Jewish groups have abandoned the rules relating to kohen marriages). In that case, it would restrict somewhat her dating/marriage options.