That's not a socialist position. To be a socialist and a man is to be absolutely committed to ending the oppression of women. It is think that you have a responsibility to learn everything there is to know about how women are oppressed by listening and immersing yourself in the history of struggle. To be a socialist and a man is to believe, first of all, that you have a responsibility to refuse participate in the oppression of women. Second of all it is to believe that you have a responsibility to be an active participant in the struggle for women's liberation. You have to think that the fight for women's liberation is your fight too. The same goes for straight people and the fight against lgbt oppression, for white lgbt people in the fight against racism, and so on. To be a socialist is to think that you have a responsibility to stand in solidarity with all groups fighting against oppression, to see their struggle as your own struggle. That's the core meaning of the politics of solidarity, of uniting and fighting.
A major way that racism functions in contemporary societies is that many white people feel indifferent to the suffering of people of color. Their default position is to think that black suffering is their problem, but someone else's problem. This was obvious during the aftermath of Katrina. What this makes clear is that many whites don't see racial oppression and inequality as a problem which they have a responsibility to struggle against. We can imagine them telling themselves things like "well, that's not my problem, that's just a problem for the black community to solve." You see this same thing in the conservative victim-blaming ideologies of "self help".
Socialists completely reject this way of thinking. Solidarity requires that we care about the pain and suffering of other human beings. It means that we take the oppression of any one particular group to be an injury to all. We don't carve up society into different racial groups and say that injustices faced by one group are only of concern to members of that group. The politics of solidarity requires that you see the struggles of the oppressed as your struggle as well. It requires that you make it your business to learn about the history of resistance to all forms of oppression, even those that don't affect you directly. The more people that are enlisted in the fight against oppression the better.
Modern racism emerges out of European colonialism and the slave trade. It grows out of a need to justify the enslavement, domination and subordination of non-white peoples to the ruling classes of Europe. Thus, racism and capitalism co-originate and make each other possible. Modern capitalism comes of age in the context of expropriations of indigenous populations, colonial extraction of natural resources, and the enslavement of human beings. It is for this reason, that race and class are deeply intertwined. The class system has always depended upon racial oppression, and racial oppression has always occurred in the context of class divisions. Thus, to think that you can overthrow one without the other is naive. For example, the Latino and Black liberation struggles in the US, in periods of heightened struggle and radicalization, have always concluded that fully abolishing racism means doing away with capitalism. Read the 13 point program of the Young Lords Party, or the program of the Black Panthers. They are uncompromising anti-capitalists. I think that's the kind of perspective the Left needs today.
Black oppression is not entirely reducible to class oppression, but it is inextricably bound up with it. Capitalism comes into existence "dripping", as Marx vividly put it, "from head to toe, from every pore, with blood and dirt". That is to say, Marx's theory of primitive accumulation had it that capitalism only comes to exist on the back of the dispossession of peasants, the expropriation of indigenous populations, colonialism and slavery. Racism and capitalism are linked from the very beginning.
This view, that racial oppression is not wholly reducible to class exploitation is basically the position defended by Trotsky in debates with Afro-Trinidadian Marxist CLR James within the SWP in the 30s and 40s. Trotsky's position was to reject the nationalist approach staked out by the CP as mechanical and inflexible. Whether socialists should support the self-determination line is a question of whether the masses of black people are demanding it. But Trotsky wanted to walk a fine line here, he did not want to simply reject the call for self-determination out of hand. On the contrary, Trotsky sensed some latent racism amongst leftists who decried self-determination because it "distracted from class". Trotsky said of this phenomenon that "the argument that the slogan for self-determination leads away from the class point of view is an adaptation of the ideology of the white workers". "The Negro", Trotsky argued in 1939, "can be developed to the class point of view only when the white worker is educated", i.e. only when white workers are disabused of racist beliefs, when racism is smashed within the labor movement. For Trotsky, however, the black struggle against racism should not wait for white workers to be won over to anti-racism, it had to begin immediately, and the job of all socialists was to support such struggles in whatever form they took. He thus argued for a "merciless struggle against... the colossal prejudices of white workers [which] makes no concession to them whatsoever".
What is the relationship between race and class? In order to answer this question, we have to talk about social relations, the level of development of the productive forces, the mode of production, distributions of power, geography, culture, the basic structure of political institutions, in short all the things that make up the bread and butter of the Marxist analysis of society and history. That is to say, race must be understood materially and historically, i.e. in terms of the material conditions of society in its historical context.
The material character of racism makes clear why we can't disentangle race and class. This is why we cannot say that race is a mere epiphenomenon that is only to be found in our language, culture and discourse. Race isn't just an idea or concept which we can critically analyze by solely examining the genealogy of its movement in thought and language. Neither is racism a merely individual or ethical problem that happens to afflict certain individuals. Racial domination is materially inscribed in the basic social institutions that constitute modern capitalist societies. It is a structural feature of the system. We therefore can't properly understand contemporary capitalism without understanding the function that race plays within it. Capitalism has always been racialized from the very beginning. But if we can't understand capitalism without understanding racism, neither can we properly understand race without understanding how racism has always been entangled in other social relations of power, in the economic structure of society, etc. That is, we can't understand what the roots of racism are unless we understand the historical development of contemporary capitalist societies (including their imperialist and colonialist projects). No critique of capitalism without the critique of racism; no critique of racism without the critique of capitalism.