It's a little bit like moving from Italian/French/Spanish to English, except that English has some tenses with no direct equivalent in those languages and a ton of phrasal verbs to learn, but that's vocabulary and not grammar.
Also, possessive pronouns are exactly like in English, concording in gender with the owner, not the object. Some people can't wrap their head around that it can be the other way around, e.g. Italian "sua madre/suo padre" can mean both his and her mother/father. In German, they must concord with both, sein Vater, seine Mutter, ihrer Vater, ihre Mutter. But Russian regional dialects do have the same feature, and if your teacher isn't a mad purist, they can easily give examples: евойная, еёйный.
Otherwise, indeed, there are less features. And in Indo-European, they're all the same: compound past tense, participles, compound past and future.
To give an example of another system: Turkic languages. 4 modal verbs (to run, to walk, to stand, to lay down), that must be applied to everything except the verb "to be", they indicate how much hurry you have doing what you're doing. It's a bit similar to Russian aspect (complete/incomplete), but way more complex. Plus you have noun cases, and everything is a suffix, and the verb is always the last. So, "I don't do X" will be something like "I <verb+ing> <stand>+me+not" (like those German prefixes that fall down in the end of the sentence.) My colleague, a Kazakh born in Russia, learns it as a foreign language, and he says it's hard.
This is only true in third person singular. For example, in first person singular: 'моя чашка' (my cup, 'cup' is feminine) vs. 'мой ключ' (my key, 'key' is masculine). Third person plural: 'ихнее дело' (their business, neuter) vs. 'ихняя забота' (their concern, feminine) although most educated Russian speakers would object to these pronouns as a bit too colloquial (although not as colloquial as 'евойная'). Same in second person singular: 'твой друг' (your friend, masculine) vs. 'твоя подруга' (your friend, feminine). In all of the examples above, the gender of the speaker/owner cannot even be determined (grammatically speaking).
About English there is a Russian saying: "in english you write Manchester but you read Liverpool"