Well, of course, The Lucy Show wasn't literally a continuation of I Love Lucy. The basic concept of the show was two women trying to get along without men in their lives, so they had to do things like put up a TV antenna on the roof and fix the plumbing and do their own finances and try to get dates, and raise their kids alone, etc.
But it's well documented in the books on Lucille Ball and Desilu that the original writers of The Lucy Show (Pugh-Martin-Davis, Carroll, Schiller & Weiskopf, who were the last team of writers for I Love Lucy) had it in their contracts that Lucy Carmichael was to have the exact same characteristics as Lucy Ricardo.
And for the first three seasons (especially the first two seasons), she did. She was clever, scheming, manipulative, bossy, childish, etc -- she even cried the way Lucy Ricardo did, and she couldn't sing well. The episodes, as in I Love Lucy, revolved around her wanting to get or accomplish something. Some episodes involved her getting involved in some wacky scheme, maybe involving disguises or lying. Others highlighted her incompetence, with everything ending up in a mess. Whatever the situation, it was just like I Love Lucy.
The books say what happened was this: The Lucy character was created by I Love Lucy's original producer and head writer, Jess Oppenheimer. He looked the other way during The Lucy Show's first two seasons, and I'm guessing this was because his old co-workers were writing the show.
But they quit at the end of the second season -- and Oppenheimer threatened to sue if he wasn't given credit and payment for the use of his character.
So Lucille Ball, who by all accounts was tight fisted and not really running things -- she did what the "men in the suits" told her to do -- rather than paying Oppenheimer, decided to revamp the show and Lucy's character the fourth season. And things were never the same again.
The Lucy character fundamentally stopped being a clever schemer and became -- well, a bumbling dumbbell. And instead of the stories revolving around her trying to accomplish something, they more often than not revolved around something happening to her -- like being drafted into the army, of all things, or becoming a super woman, or needing to go back to high school. Really stupid things, I must say.
I vastly prefer the original Lucy character. She was basically clever and quick witted, and the stories had interest and even suspense to them as we watched Lucy try to get out of the messes she created for herself. The later seasons of The Lucy Show (and the entire run of Here's Lucy) can only be enjoyed if they're not compared to I Love Lucy. Taken on their own terms, they're typical late 1960s-early 1970s sitcoms, surrealistic and kind of stupid, but with a laugh or two along the way, if you're in a stupid enough mood.