Infrared & Full Spectrum
Simply adding a filter on your digital camera's lens to shoot infrared photography can work. Digital camera sensors are much more sensitive to infrared than film used to be. The down side is that the sensor has a filter over it (called a "hot mirror") that blocks most or sometimes all infrared light. Older cameras tended have weaker IR blocking filters, so when I used the Nikon D2H (2004) and Fuji X100 (2011) it was more common that this would be successful.
This still almost certainly required long exposures (by extension; a tripod) and most likely didn't produce good results with colour infrared. As modern sensors block out more infrared light (to improve colour accuracy), this has stopped many cameras from being able to shoot infrared without conversion. For this and to allow higher quality infrared images at normal settings a conversion will likely be necessary.
One exception to this is the unusual Sigma SD Quattro, which had a removable hot mirror, making it an extremely friendly camera for infrared photography right out of the box.
Full Spectrum converted Sony RX1 + Filter set
Converting your camera to IR means removing its "hot mirror" (which blocks IR) and replacing it with an "IR pass" filter. Allowing infrared light through and blocking some or all visible light. This is an expensive conversion and one that restricts the cameras use, but it's not the only way...
"Full Spectrum" Flexibility
Instead you can have no filter on the sensor at all, a process referred to as a "Full Spectrum" conversion. This takes the camera's sensitivity from around 400-650nm up to 350-1100nm. It's important to note that whether the conversion process uses a glass layer or not is related to focusing and not protection. Find someone who is popular for doing these conversions and trust that they know what they're doing. My last two were done by Alan Burch in the UK and I am very happy.
A full spectrum conversion allows you to shoot visible light, Infrared, UV, or any combination of the three. The down side is that they will require different external filters for each combination, which is expensive and fiddly. Although it's possible to shoot UV, the filters are expensive, available light is problematic and finding the right lenses is tricky. Infrared is far more friendly, at least for exposure. Most sensors let in more IR light than the visible RGB channels combined.
Free-filtering to show the potential of Full Spectrum
Full spectrum conversions did exist first on DSLRs, but they make little sense now. In infrared the viewfinder is unusable, lenses don't focus correctly and exposure is problematic. Mirrorless cameras have none of these issues and a bunch of other advantages. You can work around some of these problems with a DSLR, so if you like a challenge and it doesn't cost you too much go for it, but for any sane infrared fans I would 100% recommend the mirrorless route for full spectrum.
The only remaining issue with a full spectrum conversion is the filters. If you have a lot of lenses this can be problematic, but there are a couple of tricks you can use to save money and complexity.
Few Large filters + Step-up filter rings
Dedicated filter holder system
Limiting lenses to one filter sizes [can combine with trick 2]
Limiting your wavelengths [can combine with trick 1 & 3]
Adapters with slot-in filters, SLR lenses to mirrorless [combine with trick 1,2 & 4]
Clip-In Camera filters
Firstly there's the simple method of buying only the biggest filters you need (usually 77mm) and then getting a set of step-up rings to adapt them to all of your smaller lenses. The down side with this is it can be bulky and your lens hoods won't work. Similarly you could buy a slot-in filter system and there you could have a dedicated hood, but this is very bulky. If you're adapting lenses (common with mirrorless) then there are some adapters that take rear-mounted filters, but this won't wonk on any of your native lenses. The logical extension of this trick is clip-in filters that go inside the camera. This is the most elegant solution. You could argue that it has the added benefit of protecting your sensor from dust, but that's only if you don't change them often. They are by far the best option for image quality as not needing external filters is always the best option, but they are not without issues. Apart from the fiddly nature of changing them (some have magnetic systems that help a bit), which exposes your sensor to scratches and more dust. The biggest issue for me was that they can still interfere with image quality. They can cause weird shadows with lens flare or they can cause internal ghosting. Due to all of the caveats mentioned above I decided to stick with the more costly brute force method of buying a ton of filters for each filter size.
Another benefit of mirrorless cameras for shooting infrared is that they allow easy old DSLR lenses to be easily adapted. These are not just cheap, but often are better for infrared due to having simpler or less coatings. This means they bounce less unintentional light around inside the camera. This effect, often called hotspot is the bane of many infrared photographers...
Sony FE 85mm f/1.8 lens - Nasty Hotspot
Infrared light contamination can subtly unbalance contrast or cause an obvious blemish. Lens coatings and lens design are the cause and digital sensors make the issue worse. I have dedicated a lot of time & money to finding which lenses work well (and which don't). I share these results free of charge here.
Special Filter - "IR Chrome"
Usually when dealing with colour infrared wavelengths (like 590nm / red filter) you're capturing a mixture of red and infrared. Some careful editing techniques can achieve a two colour blend with foliage and sky/water to look somewhat normal (like the "Switzerland" image at the top right of this page). If you want something a bit more dynamic and easier to process however, there is a great filter from Kolari Vision called the IR Chrome. Based on the famous Kodak Aerochrome film, this filter also provides some interesting foliage colour (usually orange), a naturally blue sky and a bunch of other colours. This means you get variation between the water and sky, while also yellow, greens and blues from any non-infrared reflective surfaces.