Best flight finders are Kayak and Skyscanner. Kayak is better if you have more freedom on dates but Skyscanner is good at finding low cost options.
You can get to the house by public transport but it will involve multiple changes so will take a while - car hire is the best bet if you can
Granada Airport
This is 45 mins from the house. Easyjet used to flew direct from Manchester pre covid and will hopefully resume. There is car hire etc at the airport . There is also buses via Granada to Jayena. As its only 35k so you could even ride it!
Malaga
Best range of flights but can be expensive so worth checking other options . Ryanair/Easyjet have lots of flights. Less than 2 hrs by car or 4 hrs by bus/train . Lots of car hire etc. Also always check Malaga cars and Delpaso as they often have good deals on slightly older cars, especially if you take the all in deals they offer. They have got people carriers etc's that are great with kids
Almeria
2 hrs by car or 5 hrs by bus. Ryanair are the main operators here . Lots of car hire etc. Can be quite cheap but less flights per day.
Seville
About 3hrs away by car. A good range of flights and can sometimes be cheaper.
Spending a night in Seville is a great way to see a bit more of these fantastic Spanish cities.
Gibraltar
Both about 3hrs away by car. A good range of flights and can sometimes be cheaper.