Close but not quite. To put them fairly simply:
1. Marxism: A political theory, which aims to be explanatory and predictive of how society functions, not to be confused directly with communism. So, before Marx there was a lot of other kinds of political thought and theories, but Marx was arguably most concerned with a guy named Hegel who you'll primarily hear talked about with the idea of a historical dialectic, which basically means the idea that two opposing idea come together (called the thesis and antithesis) to form a new idea, the synthesis, or the problem, the reaction to the problem, and then the solution. What Marx did was take the idea of the historical dialectic and apply it to material things, because he believed that Hegel was too abstract. He wanted to deal with the actual material world, not just ideas. He was also reacting to very common belief in a teleological process of society developing- that society ended up how it was because it was always heading towards this way, and whatever happens is unavoidable. Instead it was a series of reactions against the previous models, resulting in conflicts and new systems, which got people to where they are now (or, where they were in the 1800s). Marx then contextualized society and human history within the ideas of the means of production, and who controls them. Essentially, how does each kind of society organize itself in order to interact with and distribute material goods? When people think of Marx, they like to focus on the idea of communist revolution, which is certainly a large part of his broader philosophy, but a lot of his work was about talking about what is capitalism, and how did humans get to it, which you can separate and find value in even without buying into the idea that society should or must be moving towards communism, which is a different kind of thing (don't get me wrong, Marx believed both those things, but his analysis still had some value even without that).
2. Communism: A political ideology, ie a set of beliefs how the world should be organized. The core tenets of communism is that the means of production are owned by society, and not for the accumulation of wealth for one class or set of individuals through the exploitation of another, because there'd be no such thing as class- each person contributes what they're best at, and society gives everyone what it can. An important distinction between this and other types of socialism is the role of the state- communism, in its end form, involves the abolition of the state. Many Marxist communists would argue that a state like the Soviet Union, for example, would be called on the very extreme end of "state capitalists," because instead of the society and the means of production (machinery and industry) being owned by and for the people as a whole, it was owned and run by the state, though within Marxist Communism, state capitalism was seen as a sort of a temporary measure between capitalism and communism. Really though, if you're looking at the Soviet Union, you're not really looking at anything that theoretical communism would have really been about so it can sometimes skew people's perceptions of it- there's also the idea that an attempt for communist state would necessarily become what Stalinism or Marxism-Leninism ended up being in the Soviet Union, even if it's not what anyone but the Party wanted.
3. Socialism: Another grouping of political ideology. Communism is socialism, but not all socialism is communism. Communism is basically socialism taken to one far end of the spectrum. Socialism, at its most basic, involves some level of services being provided to all people with as close to possible as equal access. At one end, you could look at something like how health care in Canada works, and on the other you could look at communism- both are technically different types of socialism. There are softer forms of socialism where most market forces are still in play, but you have the social safety net of things like universal health care, or universal education, or whatever else, typically owned and operated by the state and funded with taxes. This is essentially what you'd see in most European countries and Canada. Every country has some level of it though, it's just a question of how far it goes and when you wanna start calling something socialist- American Medicare is a somewhat socialist aspect to your medical system, save that it only applies after a certain age.
Marxism: Political theory. An explanatory framework based on material analysis of history.
Communism: A political ideology, which can be seen as a subset of socialism. Several forms, including Marxism-Leninism, but mostly a bit more concisely defined than socialism.
Socialism: A broader spectrum of political ideologies, which can involve anything from specific ideas on specific topics (ie health care, education, etc) to democratic socialism to communism.