Hey Uncle Buck hopefully we can help.
The great thing about backyard or street cricket is of course that you can make up your own rules.
The basis of any game is simply the batter versus the bowler. So you can indeed play for hours on end with just two players.
Batting:
You seem to have the basics sorted, but just to clarify...
The batsman takes 'guard' or 'strike' (stands holding the bat ready) at the 'crease' in front of the 'wicket' (the three long sticks 'stumps' with the 'bails' (probably the single shorter stick you were talking about - in proper games it would be split in two, but in backyard sets often comes as a single piece). In your backyard game you can substitute a bin or a box or draw some stumps on a wall with chalk - anything that is about the right size will do. At it's most basic the wicket is the bowlers target (think strike zone) and the batsman is looking to 'defend his wicket' by stopping the ball hitting them. So first thing set up a 'wicket' and mark a 'crease' about 2 feet or so in front of the wicket. Easiest way to measure a crease is the length of the bat plus the length of the bat handle again. Just scratch a mark in the ground or draw a chalk line or place two rocks off to either side. This gives the batsman a good location to stand when batting so they don't hit their own wicket and is used for 'runs outs' (more on that later).
The batsman's stance is similar to baseball in they stand generally side on to the bowler (pitcher) and perhaps crouch a little, but the bat is usually tapped lightly near the feet or held slightly in the air away from the bowler but rarely higher than waist height. This is because as you noted their is no 'foul zone' and fair hits can go anywhere and because the bowlers generally bounce the ball in front of the batsman so it is usually in the batsmans favour to play a 'straight' (vertical) bat shot. Only when the ball bounces over waist height do the 'cross-bat' (horizontal) shots like a baseball swing come into play.
You score a 'run' by hitting the ball and running to the opposite crease (where the bowler bowls from). You can run back and forth as many times as you can before the fielders throw the ball back and hit the wicket. If you don't have your bat or foot placed behind the crease when that happens you are out (Run Out). The other outs are Bowled (the bowler hits the wicket with the ball), and Caught (a fielder catches a hit on the full). There are other outs like LBW but some people who've played the game all their life can't properly explain all the ins and outs of that - most people ignore in backyard games. A 'boundary' is when a hit reaches the outside edge (the boundary) of the field. If it does it on the full that is worth six runs (a Six) and if it has bounced four runs (a Four).
Still with me?
Bowling:
22 yards away from the batsmans wicket is another wicket with a crease in front of it. The strip between them is called the pitch. For backyard games you often shorten the pitch to some length that fits, but still leaves enough distance between batter and bowler to give the batsman time to see the ball. Also you can simply use one stump, a box, or just mark a crease for the bowler. So long as you have a crease the bowler as somewhere to stay behind when bowling and the batsman has somewhere to run to to score a run. You're now good to go. The bowler differs from a baseball pitcher in several important respects. Firstly in bowling the ball the bowler is meant to maintain a straight elbow. This is really hard for people who aren't familiar with the concept, best to check out cricket videos on youtube. Also the bowler doesn't start from a standing position, but can run or jog up to the crease (run up) which is pretty much required in order to bowl with a straight arm. Lastly because the wicket is lower than a baseball strike zone and you can get some advantages from the pitch the bowler is generally aiming to bounce the ball in front of the batsman. 'Full pitched' bowling bounces closer to the batsman and is aiming to hitting the wickets, 'Short pitched' bowling further away so that it bounce over the wickets but could be aiming to get the batsman out caught. A bowler will bowl an 'over' of 6 deliveries in cricket before another bowler will bowl from the other end, but in backyard games you can swap bowlers after any number you like, or simply when you think it's time for someone else to have a go. For your situation I'd suggest the option of bowling under arm (think softball). That is usually much easier for novices than bowling. It's pretty much how most people learn the game. Also Pitching the ball is fine if you want to allow it.
Fielders:
Where you have enough numbers a 'Wicket Keeper' (think catcher) is often good idea to take throws ('Returns') from the outfielders and effect a Run Out of the batsman, but plenty of backyard games do with out. In cricket a Wicket Keeper wears special padded gloves, you can go without them, or subtitute a mitt if you like. The other fielders can pretty much scatter in a backyard game, they don't wear gloves. The fine points of setting your 'Third Man' or why it can make sense to leave 'Mid Wicket' vacant don't apply. The fielder's job is simply to try and take a catch to get the batsman out, or to field the ball and return it to the Wicket Keeper. Think baseball outfielders and you'll be fine.
So that's the basics of the game. The batsman is looking to hit the ball and score runs and the bowler and fielders are looking to get them out by hitting the wicket, catching them out or running them out.
Backyard Rules:
Because a backyard game can be played by as little as two there are plenty of conventions that most backyard players will recognise. You can use any of these you like or devise your own variations to make the game fun.
Every man for himself:
Normally cricket is played by two teams of 11. Two batsmen bat at the same time and swap ends when they take a run with a new batsman coming in to bat when one gets Out. In backyard games you can simply have a single batsman who gets a free walk back to the batting end if they take a single run. You can also play with two batsmen and no teams where every batsman is playing for himself (trying to score the most runs).
Last man standing:
Similar idea, but for when you do have teams. Normally the teams innings is over when 10 wickets fall (only one not out batsman left - so if you team is 7 then after 6 wickets), in order to give everyone a fair go when you are playing in teams you would have two batsmen batting together until there is only the one batsman left. The last man standing then continue batting alone until they get out. The team is then All Out.
Hit and run:
Unlike baseball you are not forced to run just because you've hit the ball. But in order to move the game along and bring Run Outs more into play lots of backyard players play 'hit and run' ('tip and run'). That should be simple enough to understand. Anytime the batsman hits the ball with the bat they have to run and can be Run Out if the fielders can hit the Wickets with the ball before they make it safe to the other crease.
Automatic Wicky:
As you are often playing with a smaller number of players or a confined space backyard games can often feature Automatic Wicky (Wicket Keeper). Simply set up the wicket in front of a wall or fence and designate an area of that fence/wall as Automatic Wicket Keeper. Then any hits by the batsman (edges or snicks) that hit that part of the fence/wall are considered to be Caught and Out.
Six and out:
Because backyard cricket is strangely enough often played in backyards the 'boundary' is usually the fence. In order to discourage hitting the ball into neighbouring yards it is often deemed that a such a hit earns the 6 runs for the boundary, but is also Out. Can also be used for hits that go onto the house roof, hit a window, or any other place you want to discourage the ball going.
Wicket taker bats:
If you aren't paying in teams a way of choosing the next batsman is wicket taker bats. Whoever gets the batsman out gets to take his place. So if the batsmen is out bowled the bowler goes in to bat, if out caught or run out the fielder who made the catch or hit the wickets goes in. This can be varied for fairness by getting the wicket taker to choose the next batsman if they have already had a turn.
One hand one bounce:
Can be used when there aren't many fielders, so a batsman is Out caught if a fielder can catch a ball on the full (one or two handed), or on the first bounce if they catch it one handed. Can also be varied so that one hand one bounce rules applies to hits that bounce off objects in the backyard but not the ground. For example a hit that bounces off a car or the house can be caught Out but only if it is caught one handed.
No Goldens:
In cricket to be Out without scoring is called a 'Duck'. To be Out first ball without scoring is a 'Golden Duck'. It is common to allow players a life in backyard games if they are out first ball (or before they've scored) so 'No Goldens' or 'No Ducks'.
Hope the essay hasn't been too confusing. The thing is there is no right or wrong way to play backyard cricket. Just get a group together and have a hit. Fielding with a can of beer in your hand is perfectly legitimate at barbecues, letting the dog catch you out first bounce is a good rule, and making the batsmen field any balls that go under the house or over the fence is really just good backyard sense. Hope you have heaps of fun and one day we can coach you in the finer points of taping up a tennis ball.
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