Say I own some shares in a company and their stock rises and I want to buy more shares does my older price change to the new one?

Just because you buy shares does not "lock in" the price. The shares are worth whatever the market currently values them at. A stock "rising in price" intrinsically means that it now costs more to purchase, unless I'm misunderstanding your question...

Say you buy two shares at $100 each. Your position is currently worth $200. The stock price then rises to $120. You now own two shares, priced at $120 each, for a total of $240. If you buy one more share at this new price, you own three shares, each worth $120, for a total of $360. Note that in this example your position is worth $360, but you only paid $100 x 2 + $120 = $320.

I would love it if you could explain it with an example.

Say you go to the store and buy 10 apples for $1 each. You then go to the store the next day and the price has risen to $2. How much would you expect to pay for 10 more apples?

Obviously you'd pay $2 each. What you paid for apples yesterday is irrelevant.

Now suppose you buy the 10 extra apples, and and want to sell 10 for $2 (we're ignoring bid/ask spreads in this example). So you sell 10 apples for $2 each. You have 10 apples that you spent a net $10 for (the $30 you spent less the $20 you got for selling the 10). It doesn't really matter (at this point) which 10 apples you sold, except possibly for tax purposes, which I don't think you're asking about.

Keeping it simple, buying stock (shares) is a risk. The value of the stock you are holding will most likely change after buying them. If you want to buy more shares at a later date and the price has changed, either up or down, you buy at that new price, no matter what the 'original' price was. So if you sell the same number of shares and the value has gone down you receive less than what you paid originally (you have lost some money). Conversely, if the value has gone up you receive more than what you originally paid (you have made some money).