Welcome to PF.
The K-3 is a very capable and durable camera. You've made a good choice.
You mention the DA 50mm f1.8. It's a good lens for portraits and low light, but the field of view will often be a bit narrow for your intended use.
I agree with other posters that, given the limited budget and the range of requirements, a zoom would be a better option to start with - the obvious choices are the Tamron 17-50mm f2.8, a Sigma 17-70mm f2.8-4 or a Pentax DA 18-135mm f3.5-5.6. I'd suggest the Pentax for these reasons:
- it's weather-resistant, unlike the other two;
- the wide zoom range makes it very versatile (e.g. kids and pets at a distance one minute, up close the next);
- centre sharpness is excellent and the colours are very good.
For real-world samples see this thread:
DA 18-135 WR, Show us what it can do - PentaxForums.com You should get a good copy for around $US200 second hand. Look in the marketplace here:
The Pentax Marketplace | Buy & Sell Pentax Cameras and Lenses - PentaxForums.com
That leaves about $150. You will probably want something with more reach for wildlife, so that probably means an xx-300 zoom. Norm has mentioned the Sigma 70-300. Another to consider is the Pentax DA or DA-L 55-300mm f4-5.8. You should find one of them for around $100-120. (There is a weather-resistant/HD model too, using the same optical formula, but it would be more like $250 second hand.) Down the track, if the budget is available, you might want to upgrade to the current model, DA 55-300mm f4.5-6.3 PLM - which has very quiet and fast autofocus (the older ones are noisy and rather slow), but the older ones will suffice in the meantime. The older models can produce sharp images (especially if you can stop down the aperture by one or two stops). Here are some examples with the DA-L 55-300mm f4-5.8:
You can't really ask more from a $100 lens. More samples here:
Pentax DA-L 55-300mm f4-5.8 sample images - Des(Australia) - Flickr
I would strongly suggest you keep a little of your budget for a flash. Yes you can use the popup on the K-3, but you will get much much better results with an external flash. Look for one that has a swivel head, so you can bounce the beam off walls and ceilings when the flash is mounted on the camera. You can get powerful manual flashes (ie you set the flash strength manually) for around $60, but I think you would find an automatic flash, using Pentax through-the-lens (PTTL) metering, much more convenient.
There are good third-party PTTL models (Metz, Godox, etc) which would be quite satisfactory on-camera, but for off-camera use you need a way to trigger them (another flash or a radio trigger) - on a tight budget, it may be better to get a model that can be triggered off-camera out of the box. The Pentax models can be triggered off-camera by the popup flash on the K-3 (with or without the popup flash contributing to the lighting of the image). I'd suggest you look for a second-hand Pentax AF-360FGZ II. Or if that is too expensive version I of that flash would be a possibility but its capability for on-camera use is limited because it doesn't swivel.
An alternative would be to get two Godox TT350P flashes (one on-camera to trigger the other off-camera) or one TT350P and one Godox X-Pro P trigger. The flashes are about $US75 each and the X-ProP about $US60. The Godox units are very compact, light-weight, well featured (they include HSS) and easy to use. The only downside is that the recycle time is rather slow, because they run on only two AA batteries.
For more information about flashes for the Pentax system, see this guide:
Comprehensive Pentax Flash Guide - Articles and Tips | PentaxForums.com
It's really worth learning about using flash effectively. Just the simple expedient of bouncing the flash from more surfaces will make a big difference to the quality of the exposure. Bouncing diffuses the light and gives much more flattering results. Have a look at these articles from Neil van Niekerk:
Bounce flash photography tutorial bounce flash photography - be bold! How to bounce your flash
In this video, van Niekerk tests a lot of diffusers for an indoor on-camera flash. The best results? No diffuser, just bouncing the bare flash pointed behind him.
Better still, use directional bounced flash with a flag. Van Niekerk demonstrates this with his "Black Foamie Thing" (a piece of black card strapped to the flash) in the last couple of minutes of the video. See also
Indoor Exposure Mixing - PentaxForums.com
Flash can also be a big help with outdoor shooting too, e.g. to brighten up shadows on faces in harsh sunlight. (A flash with HSS helps for this.)
You may also be surprised at how useful fill flash can be when photographing wildlife, to bring out details. Here's an example.
The key is not to overdo it. Just aim for a one-two stop increase in exposure.
My other tip for a beginner would be to take photographs in RAW + jpg formats. Jpg will give you shots to view and share immediately, but RAW images let you make a lot more adjustments. There is a learning curve with post-processing but it's well worth it. If you aren't up to learning to use processing software straight away, save the RAW files for the day when you are. My biggest photographic regret is not shooting RAW for the first 6 years in which I had a DSLR - I look now at the jpgs and think how much more I could have got from those images. For images of your children, you would hate to be saying the same thing years later. Store the files and back them up (either in the cloud or to an external hard drive stored somewhere else) to preserve them.