588 out of 720 in Neet 2020 means your expected rank would be around 10000 to 12000 and the expected percentile would be around 99.00 to 99.50. Now the score depends on various factors and the cutoffs do change every year. So it totally depends on what State you want to apply and if you want to apply under All India Quota or State Quota. If w ego through the cutoffs of various colleges, anything less than 550 for General category is less. There are few states like Tamil Nadu, Karnataka, Uttar Pradesh, West Bengal etc have higher cutoffs while states like Himachal, Assam, Bihar, Maharashtra have quite less cutoffs. And you cannot apply for State Quota of every single state which means you should also go for All India Quota. However, in general, 588 in Neet is quite a good score. You can expect a good government medical colleges in your State quota such as Maharani Laxmibai Medical College, Uttar Pradesh University of Medical Sciences, GMC Badaun, GMC Kannauj, GMC Saharanpur etc. You can check the cutoffs of Various Medical Colleges of Uttar Pradesh http://medicine.careers360.com/articles/neet-cutoff-up
I would suggest you to use our Neet college Predictor tool to know where you have good chances http://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer
588 is definitely a good score to fetch you a seat in government college. Even if you try for All India quota, you have fair chances of getting a government seat.