Most mistakes that people make when speaking a foreign language are a result of what linguists call "transfer" from the native tongue to the target language. Most mistakes are logical and predictable (if you know the person's native language well enough). English people will say пИсать instead of писАть in Russian for instance.
..and will make mistakes with "если" (if) constructions as in: "я не знаю если он придет сегодня или нет" instead of "...придет ли он..." - this second example is transfer.
Some mistakes by Russians include:
- I feel myself good 
(should be simply "I feel good" - without "self")...from the Russian чувствовать себя. In English this sounds strange because the verb "to feel" has both an emotional as well as a physical meaning. So, "feel myself" is like saying "touch myself"! Трогать or щупать yourself! Звучит не очень прилично короче. Так что лучше без "self" - eg. "I feel happy, sad, annoyed, depressed, good, bad, etc".
- Do a mistake (should be "make a mistake")...the problem here is that in English we use "make" and "do" where in Russian you mostly have делать. Generally speaking, "make" is closer to создать - when you produce or create something...it could be a cup of tea or it might be a mistake - doesn't matter, it's still "make". "Do" is more about work, tasks, routine processes...for instance: do homework, housework, a task, the washing-up, a presentation. So you could "do the cooking" but "make a meal" - the first is focusing on the process, the second on the creation of the meal. 
The distinction is not always this concrete - there are exceptions to the rule.
- The bird is on the tree (should be "in the tree")...mostly we agree on prepositions in English and Russian...we have categories such as "surfaces" or "technology" (on or на), "inside" (in or в), and so on...but there are exceptions and this is one. In English we say "in the tree" as if the leaves represent a 3D object with an inside...in Russian perhaps you tend to imagine a tree as being made of branches - thus "on the tree" or "на дереве" would be more logical...in English we also agree that "branch" or "ветка" takes "on/на"...surface relationship. This is perhaps a difference in the cultural perception of trees...when you ask English children to draw a tree they draw a cloud with a stick coming down from it...when i asked Russians to draw "дерево" i often got a tree with visible branches. Maybe this is not pure coincidence. Just my idea. But, there is another interesting example like this with "picture" or "картина" - in English we have "in the picture" where as in Russian you have "на картине"...this extends to picture-like things such as "photograph" (on) and "фотография" (на).
And sometimes there are coincidences - in English we say "who knows?" and in Russian "ху_ knows". 