To what extent are the stories of The Walking Dead game series and the Walking Dead TV series connected?
The Walking Dead Season 1 and The Walking Dead Season 2 games are separate stories to the TV series. Other than sharing the same zombie-infested world, there are no spoilers between each except for possibly 2 minor details:
Glenn, from the TV series, does appear in The Walking Dead Season 1, which occurs prior to events in the TV series.
Both reveal the fact that you become a zombie after dying, regardless of being subject to a bite at different points in the series/game.
The Walking Dead Season 2 is a sequel to The Walking Dead Season 1 - you will understand the overall story by playing them in the correct order.
The Walking Dead: Michonne game is obviously based on the character from the TV series. It covers a period during the comics.
The Walking Dead: Survival Instinct is a game that acts as a prequel to the TV series (including key characters from the TV series). This is not a Telltale game.
The Escapist: The Walking Dead is a game that covers events from the comic, which is what the TV series is based upon. This is not a Telltale game.
In relation to the link between the Telltale games and the comic books, there is a question here on the site that covers the concerns:
Will playing The Walking Dead spoil the comics?
To answer your question more directly:
Should I be worried about not understanding certain aspects of the game story, or about accidentally spoiling part of the TV series?
*The Walking Dead Season 1is a self-contained story in the same "universe" as the TV series. All elements of the story are explained sufficiently enough so that you don't need to watch the TV series to understand the game. I played both seasons of the games before watching a single episode of the TV series with no problems.
I have now watched most seasons in the TV series, and can say that nothing in the game has spoiled my experience except for the fact I already knew (from the games) that when someone dies they come back as a zombie (i.e. don't need to suffer from a bite).
Everything else regarding the zombies (e.g. aiming for the head, suffering from a bite etc) are all generic traits in any zombie story anyway.