The error says it all, you're not grouping by MEMBERS.MEMBER_ID and MEMBERS.MEMBER_NAME.
SELECT MEMBERS.MEMBER_ID, MEMBERS.MEMBER_NAME
, COUNT(personal_training_sessions.session_id)
FROM MEMBERS
JOIN personal_training_sessions
ON personal_training_sessions.member_id = members.member_id
GROUP BY MEMBERS.MEMBER_ID, MEMBERS.MEMBER_NAME
You want the count of personal sessions per member, so you need to group by the member information.
The basic (of course it can get a lot more complex) GROUP BY, SELECT query is:
SELECT <column 1>, <column n>
, <aggregate function 1>, <aggregate function n>
FROM <table_name>
GROUP BY <column 1>, <column n>
An aggregate function being, as Ken White says, something like MIN(), MAX(), COUNT() etc. You GROUP BY all the columns that are not aggregated.
This will only work as intended if your MEMBERS table is unique on MEMBER_ID, but based on your query I suspect it is. To clarify what I mean, if your table is not unique on MEMBER_ID then you're not counting the number of sessions per MEMBER_ID but the number of sessions per MEMBER_ID and per MEMBER_NAME. If they're in a 1:1 relationship then it's effectively the same thing but if you can have multiple MEMBER_NAMEs per MEMBER_ID then it's not.