I understand you are on a budget, but if you want to make sure you will be protected from very cold temperatures or rain, you'll need to invest a little more into your gear than you normally would. It would be different it was not going to be really cold and it would not rain.

If you do not know the temperature rating on your sleeping bag and you can't somehow look it up, I would highly recommend buying a new bag. You'll have a brand new bag, and the comfort of knowing the temp rating so you wont have to worry about it at night.

If you are expecting rain, a tent is a must. You do not have to buy an expensive tent, just one that can shield you from rain. It will also keep you warmer (since the rain can be cold and make you wet).

I'll leave a bunch of links of some products that you can get ideas from. I see a lot of people talk about just visiting REI, they sell some good quality gear too.

I hope you find these tips useful! Also on the topic of spring gear, I would also recommend some bug spray and Sunscreen if your going hiking in Yellowstone and the Grand Tetons especially. The Tetons are at a pretty high altitude, and if its sunny you might get sunburnt even if its cold. You can invest in that or a nice hat that can at least shield your eyes and head. Ill leave some links for those too.

DICK's Sporting Goods Sleeping Bag Link

Coleman Sundome Tent (Sold on Amazon) Link

Organic Bug Spray (Sold on Amazon) Link this one is only $5, perfect for your budget

Caramber UV-Protecting Sun Hat Link

Coppterone Sunscreen (Sold by Amazon) Link

I am recommending a lot of Amazon listings because you can get some pretty good quality stuff at a cheaper price than you normally would at a retail store, as long as you have the time to wait for the product to ship to you. Again, hope this helps!