The vacancies are announced at different times. The process of recruitment and final appointment also has the time factor. So you may apply to the vacancies and job announcements as they come. You may accept the first one that selected you. Then you will feel less pressed and can make your choice comparing the various factors like remuneration, place of appointment, transfer prospects, career prospects etc.
Both banking sector jobs and government jobs are good. Please read articles and other information about the recruiting procedures and tests conducted for both and prepare for them. It is announced in media about an IBPS common examination for banking jobs and they are tentatively to start online application by first week of August. Please visit IBPS web site for more.