Depends on the type of RER Train (I take it we are still talking RER from Paris, not TGV from CDG)
The bilevel trains have a small free space at the stairs. This area does get busy during peak periods however. You have to stay with your luggage.
The unilevel trains have a small storage area at the back of each car, where you can sit and keep some of your luggage with you.
Frankly, I shudder at the thought of taking two large suitcases and a backpack. If you have any way to pack lighter, do so. Most people find it difficult to navigate with that much luggage around Europe. Hotel rooms and private cars are far smaller than in America, and people tend to rely on public transport.
I have often travelled from Paris to DLRP with my luggage and work supplies with no issues at all. But I cannot imagine getting through the turnstiles with two suitcases and a backpack, and then trying to get down the escalator and on board the train.
At peak rush hour in the downtown stations, the RER A train is fully packed. There is no room for people, let alone luggage. If you do want to travel with all your luggage, plan to do so at off peak times. Remember that peak includes the lunch period, and that in France one starts work later than in America and finishes work later. Empty seats are very rare until you reach the suburbs, and you will have a lot of black looks if you try and use seats for your luggage.