Firstly, go and read this answer to understand the difference between a "layover" and a "stopover".
For international flights, breaks between flights of up to 24 hours are generally considered layovers (although the exact rules vary from airline to airline), and thus are generally not charged extra.
Anything over 24 hours in a location is a stop-over, and generally you will be charged extra for a stopover. Occasionally fares will allow for free stopovers but normally this will only be on more expensive fares.
So it really depends on how long you want to spend in a location. If you're happy with a <24 hour stop, then the best way is to try and find flights that give as long a "layover" as possible, without going over the 24 hour mark. eg, if you can find a flight that arrives at 7am, and a corresponding departure the next day at 6:30am then you'll be good! Often the best way is to simply use a "multi-city" search on your favorite website, and set the departure date one day later than the arrival date.
Sometimes it's worth trying on multiple websites, as various combinations of website + airline may not correctly price the connecting flight - at least not on the initial screen. Sometimes if you click through on a specific itinerary it will re-price to flights to be cheaper once it realized that it's a valid connection (<24 hours) and not a stopover.
If you want to stay more than 24 hours then odds are the price is going to go up, but sometimes not by much. The price for a stopover could be anywhere from $50 up to several hundred or more, depending on the specific fares available. Once again, the best way to find them is simply to do a multi-city search and check the results.
As far as checked baggage, again it varies between airlines, but normally if your break in flight is overnight (regardless of whether it's technically a layover or a stopover) then you will need to collect your bags at the intermediate airport. Presuming you are actually staying overnight, this is probably a good thing!
Using <24 hour connections is a great way to get a cheap quick view of a city. I have been doing this for several years, and have been to many cities that I would otherwise not have been able to visit. Staying for less than one day obviously isn't generally going to give you enough time to fully appreciate a city, but it'll give you a good quick view, and is potentially enough time to decide if you want to come back later.
I've spent anywhere between about 12 hours (eg, arrive 7am, depart 7pm) in cities, right up to 23 hours and 30 minutes for a recent trip to Amsterdam - generally all at no extra cost!
