Will the price generally stay the same or can I expect pretty big profits?
One tack to take is to use the history feature of the stock info site you use, Mine will give me 20 years, if the company has been around that long.
Consider WalMart, WMT. Back in 1989 at this time you could get it for about $5 a share, Around 10 years ago it was running around 70. Today it is a little over 50.
Consider GE. In 2000 it was going for about 60, In 1989 it was closer to WalMart's price back then, but GE had some difficulties with some directions they had taken in diversification. GE, btw, was early on the Dow Jones Industrials list.
Next consider Altria, MO (it used to be Philip Morris Tobacco). In late 2006 it was selling for around 25 or 26, but 20 years ago $4 was good.
Finally, among several you might have considered Citigroup, C. Right now it is about $3.50-ish, but despite being no stranger to prices in the 50s, 20 years ago you could have got it for something closer to $2.
These have had good years, and truthfully aweful years. What they've done over the past 20 or even 30 proves nothing. But I do know this, these three things will help a lot: (1) if the company fairly consistently makes a profit, (2) has reasonable prospect to continue making a profit in the future, and (3) try to find a company that has as much or more equity than debt and looks like they will keep it that way.
As is often noted, you buy a stock, you don't marry it. So while constant trading is usually senseless, so is holding forever because of your hopes for it.