I wouldn't recommend streaming over 1080p especially if you are doing this as a hobbyist.
Some budget recommendations for low to mid tier video capture devices are:
Low tier
$20 to $100 bucks (all of these are external)
$17.99 on amazon cheap but does not allow passthrough. You will have to play on your PC monitor.
$34.99 on amazon. Offers cheap pass through capabilities
$39.99 on amazon. Same as above
$69.99 on amazon. It's older but it is at a great price right now
Webcams - IMO the most important thing here is a light source. You can have the best webcam in the world but if you are in a dark room your face stream is going to look bad.
Razer Kio on amazon. This will give you decent stream quality and offers lighting that you can adjust. Its also on sale right now for $49.99
Also,
Sleepwalker
is spot on about the mic. You can find some good quality condenser USB mics. Yeti makes some good ones and I really like the elgato wave 3 (but it is a bit pricy)
Sticking with Razer: The Seiren mic isn't bad and its low profile.
Hyper X combo: A cheap little combo on sale right now that will get you a mid tier mic and lower tier set of headphones.