help to grep/print value of a key in json that is stored in a variable?

You want the "value" from (the first object in the "dates" array) from (the first object in the "coordinates" array) from (the first object in the "data" array)

$ a='{"version":"3.0","user":"unknown_unknown","dateGenerated":"2020-07-08T11:53:23Z","status":"OK","data":[{"parameter":"t_2m:C","coordinates":[{"lat":39.23054,"lon":9.11917,"dates":[{"date":"2020-07-08T15:53:23Z","value":25.1}]}]}]}'
$ echo "$a" | jq -r '.data[0].coordinates[0].dates[0].value'
25.1

You may also want to consider installing gron, which is a utility to "Make JSON greppable!".

With a defined as in your question,

echo $a | gron

returns

json.data = [];
json.data[0] = {};
json.data[0].coordinates = [];
json.data[0].coordinates[0] = {};
json.data[0].coordinates[0].dates = [];
json.data[0].coordinates[0].dates[0] = {};
json.data[0].coordinates[0].dates[0].date = "2020-07-08T15:53:23Z";
json.data[0].coordinates[0].dates[0].value = 25.1;
json.data[0].coordinates[0].lat = 39.23054;
json.data[0].coordinates[0].lon = 9.11917;
json.data[0].parameter = "t_2m:C";
json.dateGenerated = "2020-07-08T11:53:23Z";
json.status = "OK";
json.user = "unknown_unknown";
json.version = "3.0";

which does indeed make it easy to process:

$ echo $a | gron | sed -n '/value/{s/.* //; s/;//; p;}'
25.1

$ echo $a | gron | awk '/value/ {sub(/;/,""); print $NF;}'
25.1

Using Miller, with its default JSON array flattening:

$ mlr --ijson --onidx cut -f 'data:0:coordinates:0:dates:0:value' <<<"$a"
25.1

or regex-matching the flattened element name

$ mlr --ijson --onidx cut -r -f 'value$' <<<"$a"
25.1