Below is a brief compilation of some of the more commonly-used websites and software for speedrunning and streaming. This is not meant to be entirely comprehensive, and is mostly based on my own personal experience with the software. Your experience may vary (and usually for the better).
Streaming software, timers, video capture software
SpeedRunsLive’s tools page is perhaps the best resource available for finding speedrun-related utilities. This page has links to the most common/best streaming software, timers, video game capture software, and more specific speedrun-related utilities. Check there first for your gaming needs.
https://www.speedrunslive.com/tools/
Chat viewer
I use the Chat Game Fontificator, created and maintained by GlitchCog. This is a rather incredible piece of software. It is lightweight, easy to set up, and heavily customizable. It comes equipped with a vast selection of game fonts and game borders, and supports custom fonts and borders. You can choose the rate at which the characters are drawn to the window, badge options, and additional moderation options. Best of all, GlitchCog is very responsive to feedback.
http://glitchcog.com/cgf/
Chat Bot
There are quite a few different options for chat bots, including several custom bots that some users have made. These are the three most widely-used bots, and the three that I also have used myself for some amount of time.
Streamlabs Chatbot
This is the bot I personally use. It is extremely easy to set up and create custom commands, and it shows up in your chat under a Twitch account of your choosing. It is a very good basic bot if you have no need for more involved logic-based commands, but it suffers immensely from feature bloat. Core functionality is sometimes broken for several updates while trivial features are pushed out.
https://streamlabs.com/chatbot
Nightbot
Nightbot is one more of NightDev‘s tremendous streamer-friendly utilities. Nightbot has all of the basic functionality that any good bot should have, and is incredibly stable from my experience.
https://beta.nightbot.tv/
Moobot
Moobot is another very popular bot. I have personally not had a good experience with it, but many others have had great success with it. Definitely worth trying if you’re selective about your bot capabilities.
https://twitch.moobot.tv/
Alerts
Streamlabs
I use Streamlabs, formerly known as TwitchAlerts. Very versatile and intuitive program to set up. This can manage alerts for new and returning subscribers, tips, bits, follows, and hosts. The desktop app also locally updates text files for new subscribers, total subscriber count, highest tip of the stream session, and dozens of other stats that may be of use to a streamer. The only downside is that the desktop app tends to cause CPU spikes when playing alerts.
https://streamlabs.com/
Streamtip
NightDev‘s alternative for managing tips and alerts. From my experience, this is slightly more complicated to set up, but is able to produce more unique alerts if the user is dedicated.
https://streamtip.com/
SNES Input Viewer
I use a custom piece of software called “Input Control”, created by Omnigamer 
. In addition to serving as a live input display, it can also record the inputs for playback in Bizhawk.
https://sourceforge.net/projects/inputcontrol/
“Input Control” is based off of NintendoSpy. For more details on setting up the hardware (Arduino, SNES controller extension cable, etc), please visit jaburn’s NintendoSpy page:
Other speedrunning sites
There are quite a few websites out there for speedrunning-related content, but these are the ones I would recommend checking for more game/run information and discovery of new runners.
Speedrun.com
This is by far the most popular leaderboard out there. The quality of the runs/leaderboards is highly variable, and there are no overall rules that SRC enforces for all games. Many of the leaderboards are held to a very high standard, use consistent rules, and have well-maintained resources. Many others are the opposite. It is important to not take the listings here as law, but it is a good starting point if you are curious about a game.
https://www.speedrun.com/
SpeedRunsLive
SRL is the go-to place for speedrun races, and has been for many years. The front page is also sometimes a good place to browse and discover new runners. While not quite as popular as it was before, it is still the best place to find, watch, or participate in races.
https://www.speedrunslive.com/
SpeedDemosArchive
SDA is as old-school as it gets. While it is mostly obsolete as far as records go, it is still an important place for historical information on runs. For years, SDA was the central place for speedrun discussion, and all the old discussions in the forum are still archived. If your game has a thread or a run on SDA, there is likely a treasure trove of information to be found.
https://speeddemosarchive.com/
TASVideos
This is the home of all things tool-assisted. Whether you are looking for the most recent/fastest tool-assisted speedruns of your favorite games, information/discussions about a speedrun you are working on, or the latest and most accurate emulators for practice, TASVideos is the right place to go. The forums and “game resources” pages have an unbelievable amount of information and are invaluable resources for speedrunning.
https://tasvideos.org/
Mega Man RTA leaderboards
The Mega Man community prefers to maintain their own leaderboards as opposed to using speedrun.com, and they do an excellent job of it. If you are looking for any information regarding a run of any Mega Man game, this is the first place to check.
https://megamanleaderboards.net/
