The basics are Spark+Compression+Fuel=a running engine - there are a few exceptions but this covers 95% of the battle - check the compression - if you have no gauge then lift the saw, hold it by the starter handle and drop it - the saw should drop slowly downwards with the compression stroke slowing it up - 10 - 12 seconds is fairly normal for full extension.
Take the muffler off and check for a scored piston - 9 out of 10 saws I work on have been seized - partial seizure can occur - a slight pick up on the piston holding the ring in.
Fuel - make sure the adjusters are one turn out
If the saw is a pig to start, after 5-8 pulls, is the plug wet or dry - wet is good, dry and I would suggest a strip down of the carb and blasting it with carb cleaner - check the filters, impulse port/line and fuel line and tank vent.
Make sure the fast idle throttle lock is engaging and staying on when you try to start the saw.
IMO - dry the saw out, remove the plug and turn the saw upside down and pull the handle fast to get rid of puddled fuel, dry the plug out and check for a decent spark, replace it and make sure the carb is set to standard settings as above, make sure fast idle and full choke is used, yank the handle 6 times, if the saw doesn't at least fire a little then remove the plug and see if it is damp, soaking or dry.
Spud