Most introductory logic textbooks that I have skimmed through in a while, keep the terms 'sentence' and 'expression' undefined.
I would intuitively see
Earth is round.
Why didn't Harry come to the party?
Come here, Harry!
x lives in Norway.
all as sentences. However, 'x lives in Norway' is usually not taken to be a sentence in logic because it contains a variable. It rather gives sentences after variables have been replaced by constants.
It seems to me that the term 'expression' has a broader meaning and sentences are special cases of expressions. But still what exactly is an expression? Would
arh ahfb hghd udh
be an expression? Would 'arh', 'udh', etc. be expressions? If 'arh ahfb hghd udh' is an expression , would it be a sentence? Is it necessary for an expression to have a meaning? All these troubles imo are stemming from a lack of definition of terms 'sentence' and 'expression'.
How exactly can we define these terms?
A preliminary observation. From the time of Aristotle, logicians have used a ‘divide and conquer’ strategy that involves introducing simplified, tightly-disciplined, languages. For Aristotle, his regimented language was a fragment of very stilted Greek; for us, our regimented languages are entirely artificial formal constructions. But either way, the plan is that we tackle a stretch of reasoning by reformulating it in a suitable regimented language with clear definitions of what count as sentences and other types of expression, with much tidier logical operators, etc., and then we can evaluate the reasoning once recast into this more well-behaved form. This way, we have a division of labour. First, we clarify the intended structure of the original argument by rendering it into an unambiguous simplified/formalized language. Second, there's the separate business of assessing the validity of the resulting regimented argument.
The point of going formal, then, is precisely to sidestep messy questions about ordinary language such as “exactly what makes for a sentence?”. For when we turn to formalized languages we can of course give precise definitions of various kinds of expressions, of which sequences of expressions constitute well-formed formulas, and of which well-formed formulas are sentences.
So short answer: logicians just don't need to worry too much e.g. about what counts as a sentence of ordinary language.