I've just started eating Paleo (pescetarian version), in an effort to both lose weight and become healthy (after years of unhealthy eating including an eating disorder). I haven't had any issues so far concerning the diet, never count calories, always feel satisfied, and I have even dropped about 7 or so pounds (though I'm sure this is mostly water weight). I'm a 5'8", 145 lb. female looking to lose excess fat and simply tone (not necessarily build "vanity muscles").
Now that I have become comfortable with the diet aspect, I'd like to incorporate exercise. I intend to walk 6-7 days of the week for anywhere between 1-2 hours per day, do Tabata sprints once or twice a week, and do bodyweight exercises or Burpees about 2 or 3 days out of the week. Would this regime be a good start in an effort to lose weight alongside eating Paleo? Should I focus on doing more Tabata sprints?
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That sounds great actually. Body weight exercises such as squats, push-ups, sit-ups, crunches, leg lifts, pull-ups, dips, planks, jump-rope and lunges are all great. Walking and sprinting are also very beneficial. My only advice would be to take it easy on the burpees because it can hurt your back, you'd probably be better off just doing the push-ups and squats (squat+jump) separately because that motion of kicking your legs back can hurt your lower back. Also go easy on the sprinting at first, I tore my right hamstring muscle once, not fun and very painful. Avoid exercises which could cause injuries since injuries will kill your motivation big time. But aside from that walking and simple body weight exercises are the way to go.
doing more than two sessions of tabatas in a week would not provide any additional benefit for what you are looking to do.
If I were you, I'd only do one tabata (on a exercise bike) or sprint session a week, and do one of your burpee sessions in "tabata style".
It seems like you are taking the "lowest total time spent exercising" route -- that's ok as long are you are otherwise very active. If not, it is more important to have an active lifestyle than it is to do high intensity exercises.