In a very compressed nutshell, modern logic is concerned with relations of implication, contradiction, independence and the like between propositions and predicates.
In contrast Hegel's logic operates not on propositions or predicates but on 'notions', Begriffe, or (roughly) concepts. On his account certain concepts are more adequate to the nature or expression of reality than others. The concept of determinate being, for example, is more adequate to, more informative about, reality than the mere concept of being; and being, distinguished as finite or infinite, is more adequate than merely determinate being.
He moves up through concepts - very many more than these - until he teaches absolute being, the whole of reality or the Absolute, than which no other concept can be more adequate to reality.
So Hegel is not doing badly something that later logicians are doing better; he is doing something different. This isn't to say that he does not use propositions or predicates to formulate his logic but his logic is not about them. Nor is it to say that his logic is without internal faults. Not all the rungs of Hegel's ladder are secure.
To stress the contrast from a different angle ...
What modern logicians are concerned to formalise are, to repeat, relations of implication, contradiction, independence and the like between propositions and predicates. Hegel's logic of concepts is off-beam from this concern, answerable as it is to the admittedly unusual idea of adequacy to reality - for which, moreover, Hegel has his own criteria. There is also the problem that it is hard to disentangle Hegel's logic of concepts from his metaphysics and epistemology. Unlike modern logicians, Hegel connected logic explicitly with metaphysics and epistemology. All three are interwoven in his philosophy. Or to change the metaphor, they make up a kind of triptych. We are apter to keep things separate. Moreover, the three comprise a philosophy of Absolute Idealism which, for good reasons and bad, is held by few philosophers nowadays and even fewer symbolic logicians.
If for this reason Hegel does not offer a logic which squares with modern logic, he also repudiates the traditional Aristotelian logic and its Scholastic outgrowths. Logicians who are sympathetic to this traditional logic find little of relevance to them in Hegel's logic. Hegel falls between two stools of ancient and modern.
You still want to read Hegel ?
Hegel's Encyclopaedia of the Philosophical Sciences covers logic, nature, and spirit ('mind') in a relatively accessible way - relative, that is, to his Science of Logic and Phenomenology of Spirit. You can pick up logic, metaphysics and epistemology from all three. The part on logic, usually called The Lesser Logic, is probably the most accessible if you're new to Hegel.
G.W.F. Hegel, The Encyclopaedia Logic, G. W. F. Hegel, T. F. Geraets (translator), W. A. Suchting (translator), H. S. Harris (translator). ISBN 10: 0872200701 / ISBN 13: 9780872200708
Published by Hackett Publishing Company 1991-10-15, Indianapolis, 1991.
Justus Hartnack, An Introduction to Hegel's Logic (Hackett Classics Series). ISBN 10: 0872204243 / ISBN 13: 9780872204249
Published by Hackett Publishing Company, Inc.,1998.