Query about the expression: けさ 、 わがや で はつ ご おり を かんそく し ました

What does means that “ご” and that “し” in the end of the expression, followed by “ました”?

You should think of はつごおり as one word. It is an error to write it はつ ご おり with two breaks in it.
The word comes from はつ (初; first) + こおり(氷; ice), so はつごおり is “first ice (of the season)”. No separate ご in there.

It seems this kind of wrong breakage happens often. If you do not realize it, you’ll be lost on the wrong path.
To avoid such misunderstanding, you can always look at the kanji (or even romaji) page of the text for confirmation. Even if you don’t know any kanji, you can match the particles (は、を、と、で、に, etc) to verify the breakages between the original and the kanj text. In this case, the kanji text has it as 初氷を, and romaji text has it as “hatsu-goori o”, indicating a single word.
This way you can avoid the confusion due to typos.

かんそく し ました - it is かんそく(観測; see, observe, monitor) + する + ました (する becomes し before ます or ました).
かんそく is the kanji-based stem noun, and you add する to make it a verb, so かんそくする(観測する) means “observe”.
Adding the polite form of past tense ending, it becomes 観測しました, meaning “observed (saw)” in a polite form.

So the whole sentence is “This morning, at our house we saw the first ice” (かんそく seems unnecessarily heavy word here:).