Yes, I agree, Amtrak from BWI station to Penn Station. (Amtrak does not go into Grand Central anymore.) From Penn you could walk to the Port Authority bus terminal and catch a Shortline/Coach USA bus to Highland Falls, the little town just outside USMA’s main gate. Or you could take the subway from Penn to Grand Central and then take the Metro-North commuter rail to the Garrison station just across the river from West Point. Then take a cab from Garrison to West Point.
From Annapolis to BWI you could do an airport shuttle service like Blue Van. There is also a commuter bus from Annapolis to the New Carrollton metro station where you could also pick up Amtrak, though you are starting out farther south. USNA to USMA is completely doable without a car, but it will of course take some time. Enjoy both summer programs!
ETA: I just checked USMA’s summer seminar website. Transportation is provided from Newark airport to West Point. If you time your arrival just right, arrive at Newark and catch a West Point bus. Conversely, I believe USNA offers bus transport back to BWI after their summer program. But I realize you said you have a two day gap in between. Do you have family anywhere in the area that you are spending time with? Or are you on your own and will stay in a hotel?