Yes there is racism in Israel. There is racism towards Israeli Arabs (although nothing even near to the vicious hatred Arabs outside Israel seem to have for Jews and ‘Zionists’, and it's not state racism, Arabs have full citizen rights, although there are inequalities that are, one hopes, gradually being fixed) and there is tension and some animosity between groups of Jews who came to Israel from different parts of the world. European Jews used to feel and act superior to Jews from Eastern or Arab countries.
But this has changed immensely in the last twenty years. For one thing, it is extremely “uncool” to be Ashkenazi in Israel these days. Ashkenazis may still be a relatively strong, affluent segment of the Jewish population, but Eastern Jews have become more and more prominent in politics (where being an Ashkenazi is a very clear setback), in the army, in government ministries, in education, in entertainment (where they rule supreme), and more or less everywhere else. And mixed marriages (such as mine) are gradually rendering the distinction increasingly irrelevant. I have a little story I would like to tell about this, but now is not the time.
Update: I'm not being mean not telling. Its just that I'm not in the mood. Here's me babbling on about anti this and anti that while little girls are being shot point blank in cold blood and people are getting their heads sawn off (I couldn't watch it) and others' heads are being played soccer with.