I would second the one with Jack Cassidy as a magician.
Lots of good suggestions here, thank you 
I’ll read up on suggested episodes on Columbo fan blogs to make a choice.
Just to wrap this up, the one with Ross Martin is Suitable for Framing mentioned above, and the one with Laurence Harvey is The Most Dangerous Match. I didn’t particularly like the latter either, but I think it had a nice clue in a witness’ recollection of who made the first move in a chess game in a restaurant - this allowed Columbo to conclude who was playing White in that match, which became of relevance later on.
This one is very informative. I used it to jog my memory of individual episodes when posting here.
My favorite one is Blueprint for Murder because it has the sweetest schadenfreude. When the killer opens up the car trunk and BAM! all the floodlights come on to reveal Columbo and a group of uniformed officers waiting for him, it’s just divine. That’s the one I’d recommend starting with. It also has some good comedy where Columbo tackles the Los Angeles City bureaucracy.
I can’t watch Forgotten Lady anymore since I lost my mom to early-onset Alzheimer’s. It’s a good episode, but I just can’t do it.
Despite the fact that the episode is @terentii’s favorite, I think it would be a terrible intro episode anyway.
The tried-and-true formula is enjoying the delicious shadenfreude of watching the smug, entitled, a-hole killer think they fooled Columbo and got away with the perfect murder, when actually Columbo is quietly outsmarting them at every turn. Doesn’t work nearly as well when the killer is a sympathetic figure of pity.
Also, when he tries to get in the last word with John Payne at the end, he gets oh-so-coolly outclassed by a guy who isn’t even trying to enjoy it.
I watched the whole 1970s Columbo run on Freevee (through Prime) recently and A Friend in Need had probably my favourite ending. Columbo’s gotcha moment is priceless.
Any Old Port in a Storm is also excellent; you can tell Falk and Pleasance were having a lot of fun with that one. They are both superb.
All the Robert Culp ones are good too.
Yes I agree with this, although the one with the ‘little old lady’ and the safe is a very good episode and she is most definitely sympathetic.
I know that later-era Colombo can be less good, but Columbo Goes to College (1990) is still one of my favorites in the whole series. Just of the smugness of the two murderers is enough to make it great, but you also get a delightful return of Robert Culp.
“Try and Catch Me.” The “little old lady” is Ruth Gordon.
This one is also unusual, since Ruth’s after-the-fact accomplice (Mariette Hartley) apparently goes unpunished.
I like this one too, even though Columbo nails the murderers using the same trick as in “A Friend in Need.”
We could do a whole thread on Columbo’s murderers. Robert Culp, George Hamilton, Robert Vaughn, and John Cassavetes are among the slimiest.
There’s a list that starts with “Jack” and ends with “Cassidy”.
One that deviates only slightly from the formula is the one where Martin Landau plays twins. That one was pretty good too.
Were those all murderers who commit a secondary murder to try to cover up the first murder? I remember that happened a not insignificant number of times.
Then there were the murderers who tried to knock off Columbo when they realized he was onto them. I can think of two off the top of my head-- Murder Under Glass, where Louis Jordan tries to poison Columbo with a rigged wine opener (but Columbo marked the poisoned opener, so it was really a trap he set up to bust Jordan).
In Columbo Goes to the Guillotine, Columbo allows a magician to try a guillotine trick on him, that had ‘acidentally’ killed the victim. There was a switch for ‘real blade / fake blade’ so the blade would convincingly cut through a melon, but not a real human head. The magician had reversed the labels on the switch for the original murder. Columbo switched them back. This was pretty ballsy on Columbo’s part, and not too believable, because this trap Columbo set depended on the murderer definitely attempting to murder him. If the murderer had decided, I don’t know, maybe it’s not such a good idea to try to off a police detective, and had attempted to set the switch to what he thought was ‘fake blade’, it woulda been curtains for Columbo.
Don’t forget Nicol Williamson!
I liked Laurel and Hardy!
The first two that come to mind in the category of “secondary murders” are “Negative Reaction” and “A Deadly State of Mind.” The third is “Dagger of the Mind.”
But! Can you name a killer who pulled off three murders?
In one episode?
Pretty much.
I can think of others with secondary murders, but three in one episode is gonna take some time.
BTW, I just discovered the one with the police commissioner is titled “A Friend in Deed,” not “Need.”