After I send the 3 day notice or quit when can I begin the eviction proceeding

You need an attorney... you served the notice incorrectly and there is NO 30 day involved if/when the notice is served correctly...

A 30 day notice is given only when you evict on a month to month rent for other than failure to pay rent. Unless your state requires it for this.

If your state only requires a 3 day pay or quit, then you never give a 30 day notice. Once the legal notice is given and they do not move, that is when you file for a eviction at the housing court.

Let me try to make this clear. An evistion is a formal proceeding where you go to court to get the court to order that the tenant vacate. Armed with the court order you can then get a sheriff to forcibly remove the tenant.

Before you can start eviction proceedings, you are generally required to give the tenant a written notice with a time frame telling them that they need to vacate or repair the lease violation (i.e. pay all rental and fees due).

If the tenant has not moved by the date indicated and has not repaired the lease violation, then your next step is generally to go to court.

You did right, at least in CA you serve 3 day to perf or quit, after those 3 days you can go to your attorney and start the 30 days notice (month to month rent) 60 if they had lived in your property for more than one year, is up to you if you want to keep this resident, if he/she is a good resident just you can charge the attorney fee and late fee plus the rent of course, if not, they have to leave and still have to pay you all of these, I mean if you have a good lawyer

Same case for other violation of the house rules.

Some of the information you have received is wrong. If the tenant has not paid rent, or has violated the terms of the lease in some other way, then you can give them a 3, 5, 7 or 10 day notice to perform or vacate. (The time depends on your state's laws.) If they haven't cured the lease violation after that time is up then you can immediately file a lawsuit for eviction in the county court.

If you are terminating the lease for some other reason then you must give the tenant a 30 or 60 day notice (depending on state law and the terms of the lease) that the lease is terminated and they must leave the property within that time. If they haven't left at the conclusion of that time then you can file a lawsuit for eviction.