1. Anything with less than two blades isn't going to do you much good. Look for razors with at least two blades. The more the better!
2. Use a shave gel or foam. Some people like to try and be cheap by using shower gel or soap, but you'll wind up with razor burn and/or irritation and wish that you'd spent $2 for a shaving cream instead! Try ones that are unscented and formulated for sensitive skin, in case you have any skin allergies or are prone to skin irritation.
3. Don't use regular shaving cream on private areas. Use a shaving gel specifically formulated for that area. This will help avoid skin irritation. Irritation in an area you can't scratch or easily get to will NOT be pleasant.
For the bikini area, I love to use Bikini Zone Anti-Bump Shave Gel. It smells great, glides on smoothly, and gets the job done! It's $3.42 at Walmart.
For razors, you can pretty much use anything as long as it has 2 or more blades, but at the moment, BIC Silky Touch (10 Pack, $2.96 Walmart) and Family Dollar TwinBlade Plus (5 Pack, $2.00) are my top 2 at the moment.
If you have any budget beauty products that you would like for me to review (makeup, skin care, hair care, and related items), please send your request to me at lindsi.green@gmail.com. Looking forward to hearing from you! ;)