Internet and Phone
Madeira has generally good telecommunications services: fibre is widely deployed and much of the island has good 5G coverage.
If you're visiting Madeira from the EU, you probably don't have to give much thought to your Internet and phone usage - the EU roaming rules mean you should, within reasonable limits, be able to "roam like at home". If not, you need to check with your service provider.
Check the terms of your contract to see how calls you make to numbers in Madeira are treated: because of the way roaming works, they're technically "international" calls from your home country. Some operators may treat them as such and others may make an exception for the country in which you're temporarily located. Call diversion and access to your voicemail may be more expensive while you're roaming.
If you don't have free/cheap roaming - or you anticipate using the Internet significantly more than your contract allows - you might want to get a local data SIM for the duration of your stay (unless you're making a lot of them it's unlikely to be worth getting a SIM to make local phone calls). Of course, to make use of it, you'll need a dual-SIM phone with a spare slot, an old phone with a "hotspot" function or a personal hotspot/MiFi device.
There are three network providers on Madeira - Vodafone, NOS and MEO - as well as "virtual" networks that use their infrastructure. For short-term, prepaid, Internet access, Vodafone is currently (late January 2023) by far the cheapest option at €15 for 15 days of 'unlimited' data on their Vodafone Go Total tariff. They also do bundles with MiFi devices included. Both NOS and MEO are presently charging €15 for 8 days. Of course price is not the only factor - coverage and congestion in your specific location may be a consideration - but all the networks provide reasonable coverage. Although you can top-up the cards once your time is up it is often easier to buy another one rather than wrestle with the apps and online portals.
MEO also offers vouchers to access their WiFi hotspots, but at €3 per hour or €25 a week it's a relatively expensive option.
Telecommunication prices are expected to rise on 1st March 2023.
There are Vodafone outlets at the airport, in Funchal (Madeira Shopping, Forum Madeira) and in Machico. NOS can be found in Funchal (Rua EUA, La Vie shopping centre, Anadia shopping centre, Madeira Shopping) and Caniço - though many of their locations are primarily for customer service rather than sales. MEO has shops in Câmara de Lobos, Funchal (Avenida Zarco, La Vie shopping centre, Anadia shopping centre, Madeira Shopping), Caniço and Machico.
You may also be able to buy SIM cards at branches of Worten (La Vie shopping centre in Funchal, Madeira Shopping, Cancela and Machico), FNAC (Madeira Shopping) and other mobile phone shops.