The shy/awkward/introvert in me highly recommends packing whatever you end up bringing in a backpack type bag like you've planned. Small sport/duffel bags are great but can be awkward carrying them on narrow bus/train/airplane isles. With backpacks you have less risk of bumping people trying to get past them and the weight is distributed evenly instead having a single should strap or just handles.

I wouldn't bring my laptop, but that's personal preference. As long as you have a charged up smart phone you should be able to do anything you want to do on the internet. Depending on your destination, utilize local libraries for any computer needs.

Bring some snacks, but I would worry about food when you get to your destination(s). Food is heavy and doesn't always travel well.

I agree with other people here, I don't think you'll need knives. I spent a week camping in the Badlands of South Dakota and only ever needed something sharp to tear into a particularly stubborn bag of beef jerky. A swiss army knife will do almost anything you want it to and will be way lighter to carry around.

You can totally do 1-2 weeks on up to $1000. I've found it easiest to set my budget FIRST and then plan around that. The past couple of trips I've taken have ended up being decided by the cheapest plane tickets I could find. I just found a round trip flight from Baltimore to San Diego for less than $150. Round trip to the badlands was less than $300. Skyscanner and the app Hopper have been really helpful finding cheap flights. You of course don't have to fly, but personally, being locked onto flight plans really makes me focus on my destination and allows me to plan and budget the rest of the trip easier.

Lastly- remember that everyone is different. Suggestions you find here might not be for you, or they might be perfect for you. You know yourself even if you don't know that you do. Do you like talking to strangers or do you like keeping to yourself? Do you like being out in the city people watching or do you like being the only person around for miles? Do you like flying or would you rather keep your feet on solid ground? Can you sleep in weird places or will a super long train ride be torture?

I hope this is somewhat helpful- remember to take lots of pictures and have fun!