Upon rising one should say Modeh Ani, wash 6 or 8 times w/o a beracha and use the restroom, as you mentioned. After the restroom one washes again 6 times with a beracha if you intend to daven Shacharis immediately, but otherwise it is sufficient to rinse the hands now and wash them again with a beracha later when you are ready to daven. Reishis Chochma is not usually recited by adults, but follow your own minhag.