"He would secure the house first."
Translation:Er würde zuerst das Haus sichern.
You can put it before the object or the verb, it works either way. It's a matter of focus, I'd say:
Er würde zuerst das Haus sichern and then join the search for the dog. ("He would first secure the house." Here you sort of put a bracket around the whole activity of "securing the house".)
Er würde zuerst das Haus sichern and then the garage. ("He would secure first the house, then the garage." This sounds a bit more neutral than the next one:)
Er würde das Haus zuerst sichern [and the garage only after that]. ("He would secure the house first." This sounds more like giving an opinion on whether to secure the house of the garage first: "Choosing between these two, I would first secure the house.")
Er würde das Haus zuerst sichern and then lock it. ("He would first secure the house before he does anything else with it(!).")
And you can also say: "Zuerst würde er das Haus sichern" = "First he would secure the house (and then join the search / and then the garage / and then lock it)"