Some tips: press and hold Up to jump higher than normal. All special weapons can be upgraded once (to upgrade just get a 2nd of the same weapon you are using). Abuse the save feature if you need to (to save/load press Left Trigger). My weapon of choice is the C gun, it's the first special weapon you get in the game, its next available upgrade is right at the Helicopter fight directly before he spawns in the first stage. I prefer to spray my C gun at an arc up to right usually. I also like to take most bosses apart piece by piece but sometimes it's easier to just go for the kill. You can carry more than 1 bomb at a time and it carries over to each stage. The best place I've seen to use a bomb (press X or B for Bomb) is on stage 4 (the 2nd overhead one) and the part where about 6-8 Chatter Head creatures circle around you. I think this is the shortest of the Contra games.
If you have any struggles or need advice, please reach out to me or comment and I will do everything I can to aid you and improve this guide. Contra is my jam.