How can I start analysing my games?

I'm trying to analyze my games to see what I did wrong or could be done better. Seems like I can't do so without the paid plan here on chess.com but perhaps is there a free basic tool anyone can tell me about?

You could download the pgn of your thematic games, then load it up on SCID or whatever program of your choice.  Before running it through an engine, for each critical position, type in notes regarding what you were thinking at the time, what happened, how it could've been refined.  After that, you've got something to compare against books, forums, and against engines that'll show you alternative lines from its superior calculation.  Add "edit" lines in your notes that include something new you could've done, based on what you've gathered from those external sources of information.  Compare both, what you were thinking, and what the external info says.

I probably do that more than I play the game itself at the moment.  I still suck, but I expect it to get better over time.  What do others do to analyze their own games?

I often analyze my games by putting the PGN's into Chessmaster XI and doing postmortems that way.  It does a very good job of analyzing them and telling you what you did wrong in plain language.

I've installed SCID. How do I exactly analyse? Will that tell me what moves are wrong and other possible moves I could do?

• Load/enter a game. Make sure you are at the position you want to start analysing from.
• Start an analysis engine: Tools -> Analysis Engine. Choose the engine/options you want
• In the Engine window, click the Annotate icon (immediately to the right of the variations number box)
• Choose the options you want. Wait till it's finished.

More details are available in the SCID help and on the web.

BTW, the general advice is that one gains more insight by analysing games one's self first, before using an engine.

How do I analyse myself? If I'd already knew the good moves I'd use it in the game !!! hahaha.

Not necessarily. Hindsight is a wonderful thing. You can go through and see where the game went wrong. Perhaps suggesting a few alternative moves, set a board up and play around or go through it with someone else if possible too for a different perspective.

Let's first assume that the "best" moves aren't obvious, otherwise we'd all (beginners included) be playing best moves for every move of every game, where every game leads to draw.  

Once we've conceded this, we're forced to accept control over our own destiny.

Every game analysis could be an exercise in recognizing where the "imbalances" (as Silman refers it) were, and where+when you could have exploited them with your given move in critical positions.  I concede that I probably won't consistently see the "best" moves at my level, but I probably will be able to deduce something close to it, based on board vision that I hope to attain by improving "positional" awareness.

I find that some determine "best" based on superior calculation, which is great if you can do it, but I just don't have that ability to look at deep lines like some appear to be able to so naturally.  E.g. - Many justify what's best by giving you a line, that while absolutely correct, would be a tough conclusion for me personally to arrive to.  Lines aren't by themselves an explanation, but a proof, if that makes sense.  I prefer explanations and questions that assist me in arriving to the best conclusion, and not pure answers where I have to break down the logic first.

If you've got trouble analyzing your games, perhaps you can do so by outlining your (positional) goals for a given move, what the outcomes were, versus what you could've done differently.