My baby took his first steps recently and now he is starting to walk all the time. Gloria is correct - and Travis neglected the 'starting to' part. Another option (which is again slightly different, but perhaps is close to your meaning ) My baby took his first steps recently and now he is trying to walk all the time.

I am from the United States and would change your sentence to the following: "My baby took his first steps recently and now he is walking all the time." Another acceptable form would be "now he walks all the time."

I have a feeling that "full time" is mostly used with jobs and the like. Perhaps it would be better to just say "all the time".

My baby made his first steps recently and now he walks full time. По-русски ведь тоже делают первый шаг, а не берут. Ну, и, тем самым, он начал ходить, а потому уже нет смысла повторять это словом starts - достаточно сказать, что теперь он ходит всё время. И одно незначительное замечание: обычно для определения пола всё-таки добавляют boy or girl: baby-boy (малыш) или baby-girl (малышка).

Both Gary and Travis have perfectly acceptable answers. :)