This is not a pleasant process. As you mentioned, Texas law requires a 3-day notice to pay or quit. Your next step, as owner, you can handle by yourself. While an attorney is not required, it is recommended. In Justice of the Peace court you can file forcible detainer suit and for back rent. Generally, the court date will be about 10 days out, but you normally must show proof at the time of filing that you complied with the 3-day notice requirement.
At JP court (the one for your area where the house is) you simply testify that you have not received the rent, that notice was given. Be prepared to show proof. Usually, the judge orders possession to be returned to you and the renter to move out by a certain date. Yes, it is not immediate.
At the same trial you can get an order (it's like small claims court) for the renter to pay the rent owed. Awarding possession under the forcible detainer is separate from the rent. You have the choice if he pays you whether he can keep possession or still must move out. Make sure any funds are "good funds" i.e. cash, money order, cashiers check.
Typically, the JP court appearance is enough either 1) to get the back payments, or 2) to get the renter to move out.
Never, ever, take matters into your own hands. Don't invade their space, don't remove the front door (unless it can be proved that it is in need of repair), don't threaten or monkey with them. I require all my owners to watch Pacific Heights before they start renting out a house.
Once forcible detainer has been resolved in your favor and possession is awarded to you, watch them carefully. Often revenge takes precedence. Also, as the deadline approaches for their move-out, you may have to go to the next step - eviction.
Eviction requires you to have a valid order for possession that is beyond the date. You will need to pay for the constable/sheriff to appear and serve the scofflaw. This will not be the day you come to their office, but a week or so out. At that point in time if it is not raining, you should have your hired movers there to remove the renters belongings and place them curbside. You will also have called a locksmith to change the locks. No, the sheriff doesn't supply packers or movers, you do. If it rains, you have to come back.
Some renters appeal the JP decision, which they they have the right to do. At that point, you must hire an attorney, if you didn't already use one. They will guide you through the appeal process at county court. It is rare that county reverses the JP ruling, but timing telescopes out. They may get another month of free living. If you had won a monetary award, then the renter's appeal requires posting a bond for more than your award. This usually stifles any appeals.
By the way, Texas law is different from Florida law.