You say that your RS object is in a Master-Detail relationship with Case. Assuming that RS is the detail side of the relationship, you simply need to loop over your RS records and gather the related Case Ids.
With that, you should be able to run your query with one small change.
// Declare either a set or a list to hold the Ids.
// It doesn't really matter which one, my preference is for a Set though.
Set<Id> caseIdsSet = new Set<Id>();
for(RS__c rsRec :myRsRecordList){
// I'm assuming the relationship field is Case__c
// Even though the relationship field will show you the name of the related
// record in Visualforce (guessing the same is true for Lightning), the field, in
// reality, holds the Id of the related record.
caseIdsSet.add(rsRec.Case__c);
}
List<Contact> contactsList = [SELECT Name, Id
FROM Contact
WHERE AccountId IN (SELECT AccountId
FROM Case
// This is the small change to your query
WHERE Id IN :caseIdsSet
)];
If this code is part of a trigger on your RS object, then this is the only query that you'll need. If you're not doing this as part of a trigger (or otherwise don't have access to the Trigger context variables), then you will need another query to grab the required field(s) from your object.
+edit:
So you're outside of a trigger context. In that case, there's only one way I can think of the avoid the second query; use a formula field.
Update to edit:
formula fields can't be used in semijoins. There is still merit in the example below, but to use it in your situation, you'd still likely need to query RS__c records to obtain the value of the formula field (and then you'd be using code based on my first example).
If you have a formula field that pulls down the AccountId from the Case your RS record is related to, you could use that in your semi-join like so...
List<Contact> contactsList = [SELECT Name, Id
FROM Contact
WHERE AccountId IN
(SELECT Account_Id_Formula__c
FROM RS__c
WHERE Id IN :myRSRecordsList
)
];
While the most common use of a semijoin is where you'd query the object whose Ids you're trying to obtain (i.e. Account in this case), it'll work on any object provided that you have a field that holds the appropriate Id on that object (I don't believe you're allowed any hierarchy traversal, up or down, in a semijoin).
Best Answer
Mark was right but you need to wrap the OR in parentheses.
String query = 'SELECT Id FROM Account WHERE isNew__c = true AND (Start_Date__c < TODAY OR End_Date__c < TODAY)';