If you want to start speedrunning SMB1 but don't know how to start, this is the perfect(and pretty much only) place to learn. The following was written by Wolf but I added some things to make it more up to date. This wouldn't be even close possibe without her so thank you for that. Here we go:
For starters, you'll need an emulator (assuming you're not playing on console). The overall best emulators are Nestopia and MesenRTA. Nestopia is more resource effieient and better for lower end PC's, Where as MesenRTA is more resource intensive but has more features, such as an input display and frame counter.
You'll also need a rom of the game, which we can't provide as it is illegal to share roms. You can easily find one online though. Make sure that the rom is named something along the lines of "Super Mario Bros (World)" or U instead of World. You do NOT want a rom that contains "(E)" in the name, as it is the PAL version, which is different (and bad). The CRC32 Checksum should be "3337ec46", which you can look up on this website by importing your rom. You can also compare the SHA1 Checksum (ea343f4e445a9050d4b4fbac2c77d0693b1d0922) in MesenRTA if you use that emulator.
If you're looking to play on original hardware and don't already own the console and cartridge, you can find them online on sites like Facebook Marketplace, Craigslist/Kijiji or Ebay (if you're willing to pay a bit over the market value). Do NOT use Amazon for retro stuff, as it is heavily overpriced. The average price for a console with cables, 1 controller and an SMB1 cartridge is about 80$ USD. Playing on console also means you have to use a flashcart to access practice roms or romhacks. The Everdrive N8 Pro is currently the best flashcart on the market (normal N8 also exists but it's not sold on Krizz's website anymore).
I suggest you get familiar with the levels 1st, as well as the overall physics of the game. When you feel comfortable, you can start following one of 3 Any% tutorials: the Safe Strats Tutorial by Miniland, the Fire Strats Tutorial, or Kosmic's Sub 5 Tutorial. If you're looking to learn Warpless, SuperSonic has a decent Small Fire Mario tutorial and Eddiecatgaming has a spreadsheet of the recommend route for a 19:03.
Also, you might want Pellsson's Practice Rom which lets you select a level, the framerule and make savestates, along with other cool features you might need later on. A full tutorial by Scalpel on how to use the rom can be found here Along with a tutorial by MrSwaggins that can be found Here. You can patch your rom with the ips file using this rom patcher. If you're looking to play the MWE category, you'll want this practice rom instead, which includes -1 -2 and -3 by selecting world 9.
Lastly, while this game is extremely popular, the infos about certain tricks are really hard to find, as Discord is currently our main source of knowledge database. Don't be scared to ask questions, we have a lot of knowledgeable people in this community who would be happy to help you with whatever you need. You can join the discord by using this link.
Have fun!