From what I gather (not an expert),
Pre and contemporary economists of Marx were rather close to Marx's economic writings. A lot of Adam Smith's and David Ricardo's ideas are just ported over and expanded. He shat on a lot of his contemporaries for various reasons, I think most of those you can find in theories of surplus value aka Capital Volume 4.
In general, if I were serious about understanding economics, I'd read Smith, Ricardo, Say, and other economists besides Marx, as well as Keynes and other post-Keynesians, for example, Michel Kalecki, and even those that tried to port Marxist economics to chartalism (or vice versa). I'd also read the MMT people, which are part of the chartalism school. All of these economists have become part of "heterodox economics".
Mainstream economics is basically synonymous to neoclassical economics and the bastardizations/ugly chimeras with heterodox schools. The foundational writings and writers are absolute trash. A lot of the theory seems to part from impartial or shitty axioms, or moral imperatives, and then produce a lot of theory to cope for the failings and inconsistencies of their theory. I'd say skip this part. If you want a tl;dr, I've heard the "Basic Economics" by Thomas Sowell is a good start. Since this is mainstream economics, a lot of the concepts are probably familiar with you. I got taught a course of mainstream economics in high school, for example.
As for what a Marxist's position would be? I'd think it is a pretty standard view to see Marx as the "culmination" of classical economics. And that neoclassical economics is trash. Beware though, Marx had a very sharp tongue and relentlessly criticized everything, even when the theory he is criticizing shaped his understanding about a subject matter.
As for non-marxist economists post-Marx, I'd say most of the criticism is against whatever is contemporary mainstream at that time. J is for Junk Economics for example, is a critique of contemporary neoclassical economics (I think, haven't read it).
Some economists are not marxist, but were directly influenced by Marx, such as Michel Kalecki, or Steve Keen (and other MMT folk). From what I've seen, Marxists seem to at least be tolerant to the MMT and post-keynesian theorists.
Although heated debate surely exists.
For example, Michael Roberts is a Marxist and has Steve Keen (an MMTer) linked on his blog:https://thenextrecession.wordpress.com/
And this series of articles on MMT seems to indicate he is sympathetic but mostly disagrees:https://thenextrecession.wordpress.com/2019/01/28/modern-monetary-theory-part-1-chartalism-and-marx/