Pauly - you do sound like you don't iron stuff?
Get a small folding or wooden ironing board that you can sit on your dining table. If you don't iron much consider a travel iron - but basic irons are cheap.
Washing shirts - the main danger is letting stains set on collar and armpits, washing in too hot water, thrashing around in machine and especially thrashing around in a hot dryer. Just think about it. All the stress on the material. Laundry bags help mitigate friction on shirt in washing and drying in tumbler. Tumble drying is your enemy.
To avoid stains setting - wash as soon as possible after wearing. If you cant wash straight away then soak immediately after taking shirt off. Which naturally you will do, with your dress shirt, very shortly after walking in the door and changing. Just throwing in a bucket of ordinary old water is fine. Avoid dry cleaning. Dry cleaning will set sweat etc stains.
My tip if you are living alone in an apartment is get a wide bucket - place it near shower, each night chuck your shirt into bucket full of water. When having a shower in morning - wash shirt by hand when you are in the shower. Rinse well and hang WET on hanger, plastic NOT WIRE, over bath/shower - generally by the end of the day the shirt will be dry. And with a bit of luck due to drip dry relatively free of wrinkle if you hang it right.
Every now and then soak shirts, particularity white ones, over night in Napisan or whatever they sell in USA to stop yellowing. Don't use bleach..