"Marxism" is usually understood as method of societal analysis and worldview with some core concepts (historical materialism, mode of production, class struggle, dialectical materialism) and key theories of how this society they refer to as "Capitalist mode of production" works (surplus-value, commodity production, falling rate of profit, crisis tendencies, etc). In this sense, the "end-goal" of "Marxism" is to develop a comprehensive sociological and economic understanding of capitalist society and human history.

By "Marxism" you probably refer to "communism", which is the socio-political movement that aims to create a classless, stateless and moneyless society and which largely consists of Marxists and Anarchists. In that case, there are many different interpretations of Marx and Engel's ideas on the State, developed by the social-democrats, Bolsheviks, left-communists and etc, so it's hard to say what are the "end-goals" of "Communism" because that's such a broad category. However, if you want to know what exactly Marx and Engels themselves wrote on the subject, i recommend you read this essay by the Marxist-Humanist Initiative as an introduction and Critique of the Gotha Programme and The Civil War in France by Karl Marx for the word straight out of the man himself. Another work that is usually cited as a must read is the Communist Manifesto, which is short and not particularly hard to read, but you need to know that this particular work was a political pamphlet written in a particular context (the 1848 revolutions) for a particular audience (the European working class of the time) and both Marx and Engels showed a desire to change and update the "programme" at the end of section 2 later in life.

But it goes like this: To Marx, what the working class as a whole needs to do is to smash the infrastructure of the "bourgueois State" (also referred to as the "dictatorship of the bourgueoisie") and create a "dictatorship of the proletariat". In Marxist terms, the so-called "dictatorship of the proletariat" simply means that the working class is imposing it's will upon their former dictators, this word doesn't necessarily refer to a political dictatorship or authoritarian State as we understand it today (though many of Marx's critics, particularly the anarchist Mikhail Bakunin, saw the "dictatorship of the proletariat" as a flawed and even dangerous interpretation that could be interpreted to justify a literal dictatorship). What this "dictatorship of the proletariat" does is that it dismantles the old institutions of the State (the standing army, the police, etc) while seizing the social means of production and turning them into common property for the working class as a whole, re-organizing the economy around a democratic "common plan" and substituting money for a "labor note" system to organize the distribution of resources in this new society. After old institutions have already crumbled, there is no need to repress them anymore (as they have run their "historical course" and will not assert themselves again) and the "State" (aswell as the "dictatorship of the proletariat") will have no reason to exist anymore so it will wither away.

So it's wrong to think of what Marx had in mind as a particular political party taking office or taking the Government and then implementing a specific programme. What Marx had in mind was revolutionary action of a whole social class against another, that is, everyone participating in the revolution and radically changing all social institutions. The government is substituted by the "self-government of the commune", private property becomes common property and the economy is re-organized around a democratically-determined common plan. Yet the concept of dictatorship of the proletariat has continuously been used to justify the dictatorship of a political sect, much like Marx's critics feared. Besides the fear of a dictatorship of the proletariat leading to an actual dictatorship, another "noticeable flaw" is that most modern day economists would argue that organizing an economy around a "common plan" is wasteful or problematic, however this is an economic debate that i think goes outside the scope of this post (and is another disagreement between Marx and Bakunin, as Bakunin and his followers oppose the centralisation that a "common plan" implies and propose for a "federative" or decentralized organization of Socialist society instead).