Are there any rules of thumb for when to 'walk with your chips' and realize gains made on stock?
The only general rule is "If you would buy the stock at its current price, hold and possibly buy. If you wouldn't, sell and buy something you believe in more strongly."
Note that this rule applies no matter what the stock is doing. And that it leaves out the hard work of evaluating the stock and making those decisions.
If you don't know how to do that evaluation to your own satisfaction, you probably shouldn't be buying individual stocks. Which is why I stick with index funds.
You should know when to sell your shares before you buy them. This is most easily done by placing a stop loss conditional order at the same time you place your buy order.
There are many ways to determine at what level to place your stop losses at. The easiest is to place a trailing stop loss at a percentage below the highest close price, so as the price reaches new highs the trailing stop will rise. If looking for short to medium term gains you might place your trailing stop at 10% below the highest close, whilst if you were looking for more longer term gains you should probably place a 20% trailing stop.
Another way to place your stops for short to medium term gains is to keep moving your trailing stop up to just below the last trough in an existing uptrend.
In a perfect world of random stock returns (with a drift) there is no reason to "take profit" by exiting a position because there is no reason to think price appreciation will be followed by decline.
In our imperfect world, there are many rules of thumb that occasionally work but if any one of them works consistently over a long period of time, everyone starts to practice that rule and then it stops working. Therefore, there are no such rules of thumb that work reliably and consistently over long periods of time and are expected to continue doing so. Finding such a rule is and always has been a moving target.
The rational, consistently sensible reasons to sell a stock are:
- To rebalance a portfolio toward optimal weights
- To lock in gains (or losses) for tax purposes (e.g., gains in a year when your tax rate is low or losses when your tax rate is higher).
- To cash in so you can spend your money on consumption or real (non-financial) investments.
These rules are very different from my interpretation of the "walk with your chips" behavior mentioned in your question.