That is exactly what Tapbots has done with Ivory - Taking Tweetbot and using that as the basis for a Mastodon client. I've been using it for the past few weeks and it's fantastic https://tapbots.com/ivory
Ivory from the makers of Tweetbot. The project was started before the whole Twitter API fiasco started, but apparently they've had to accelerate the development of it now.
I used to use Tweetbot as the official client is and continues to be a dumpster fire.
I then switched to Ivory and it is very much like having twitter of old back - a single chronological feed of people I follow without the involvement of any algorithms. I know that the federated bits are there, but they have very little obvious impact on my experience.
There are various tools that will mirror tweets to Mastodon. Many of them worked the other way around (replicating Mastodon posts on Twitter) but with the ridiculous pricing Twitter asks for API access I doubt they'll still work in a few months. There's also a tool that will directly mirror RSS into Mastodon, so if that's the source of your tweets then you're in luck.
There's also a decent, chance your posts are already readable on Mastodon through https://bird.makeup/. I wouldn't use that as an official source, but it's probably good to know about. It should also be noted that some servers block dedicated Twitter forwarding services like these because of the massive moderation task blindly forwarding the Twitter cesspool can cause.
I can't tell you what tool or program suits your needs most. There are systems ranging from fully custom ActivityPub servers to simple set-and-forget IFTT automations (https://ifttt.com/explore/how-to-crosspost-mastodon-twitter) that can solve the problem of "post the same stuff on two platforms". Sadly, the most easy to use tools that I know of are useless without access to the Twitter API and I doubt you'll be willing to spend the ridiculous amounts of money Twitter demands for that for a few replicated tweets.
As far as simple guides, everyone has their own idea of what "simple" means. In its most basic form, you type or paste text into an input field and hit the send button, just like on Twitter. You give people your username to follow (@user@server.com),they type that into their dashboard and they hit the follow button. If you want to reply to people, you click the reply button and type a reply. Retweeting and liking works the same way as on Twitter, with similar icons. Usernames are a bit longer because they include a domain name, but honestly who even cares about usernames on social media.
If you've ever emailed someone outside your organisation, using Mastodon shouldn't be harder than using Twitter. The hard part is synchronising the two, because the only people who have put effort into that so far either used the Twitter API or are comfortable with running shell scripts.
For a brief bit I had a simple python script dump tweets into a .plan file mostly just to amuse myself by connecting new social media to old.
Hmm, I wonder if I ask my Mastodon host nicely if they'd open up finger protocol support, lol. (ETA: Reminds one of the old adage about blind men touching the elephant.)
Another new tool in this vein is Fedifinder. It works by scanning Twitter bios of accounts you follow for strings that look like Mastodon addresses: https://fedifinder.glitch.me/
5 years ago there was a neat tool called Mastodon Bridge that did what Twitodon says it does but much better, I think because it didn't require everyone opt in. It stopped working because of some change Twitter made to their API terms of use. https://github.com/mastodon/mastodon-bridge
There's also Moa Party, but it's so complicated I have never used it. https://moa.party/
Mastodon explicitly doesn't follow these kinds of opaque algos for recommending tweets.
The risk that someone will create a "knock-off" twitter? Give me a break. Anyone could do that at any time, there's nothing special about Twitter except the enormous network effect.
The only risk I can see in open-sourcing the code is that it would make it easier for people to game the system and attack it in the short run. But opening it to scrutiny and review will make it more robust and less susceptible to attack in the medium-to-long term.
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