At least 6 months before you start applying for rentals check on your credit and solve any outstanding collection items if you owe somebody money and they send it to collections you absolutely have to get that solved and paid off. Have several months with no late payments.

Make sure you save up several hundred dollars for the application process as you may have to make multiple applications. And this might need to be more if there are two adults applying.

Always see the property prior to paying for an application

Always have the listing agent verify in writing that it's available before running your application

Always be extremely upfront with the person regarding anything in your credit or criminal past. It's very helpful to put this into writing so they know that you're serious. If you approach your issues professionally you will be dealt with on a more professional level.

Having a pet can create a lot of headwinds for you when you are looking for a property. Take steps necessary to remove or just know that you're going to need to allow yourself more time and probably more money to get the rental that's right for your family.

Consider hiring a tenant's agent. This type of agency is not terribly popular in our area however if you hire the right agent and offer them a suitable fee for service they can probably save you money in application fees and they can probably negotiate better terms than you could on your own, and most importantly it'll lower a lot of the stress that you're bound to incur.  

Be sure to document the condition of the property in writing as you move in. It's not a bad idea to take photos as well or video but written evidence is very powerful. Make sure you deliver it to the property manager in a timely manner.

I suggest renting professionally managed properties especially when those managers are members of the national association of Realtors.

Read your lease carefully before signing. No you're rights. But also know you're obligations also.

Don't plan on having your security deposit back the day that you move out as managers have several weeks to return your deposit. The better condition you leave the property in the fewer estimates they need to get.  

Paying your utilities as you're moving out and providing proof of those payments will expedite the amount of time it takes to get your security deposit returned