I am new to the photography industry, and have my first engagement booking this upcoming month. I am planning on buying the Canon T5i body, and am wondering if the EF 85mm f/1.8 USM will fit and be a good lens to buy with it.
The EF 85/1.8 should fit and will work fine for portrait, but on an APS format camera it will require relatively large shooting distance. If you plan shooting in small rooms, walls may prevent you from getting far enough from the photographed person to get enough of their bodies in the picture.
55-250 on a lower end body is definitely not the best combination for sports and action. You will have the reach, but your equipment will have problem keeping focus on quickly moving people and you may have issues in darker places. You can also use the 85mm for sports. It is not a zoom and therefore a bit limited, but it focuses fast and passes more light.
It is not a perfect solution, but you could also consider getting 70-200/4 instead of the two lenses. Unless you need very narrow depth of field of the 85mm lens. This lens will give you faster focusing than 55-250 for sport, bit more flexibility when shooting portraits and a bit of advantage in low light conditions compared to the 55-250.
The Canon T5i can use any lens that is an EF or EF-S mount. The only mount it can't use is the EF-M mount. So to answer the question of being able to use that lens on your camera, Yes you can use the EF 85mm f1.8 USM on the Canon T5i. I cant comment on the quality of the lens as I have never used the lens personally but I suspect it will be decent. You might be better off to rent it and test it before buying it to see if you like the lens or not.
Any EF or EF-S lens will fit and be fully functional on an EOS Rebel T5i. That includes the EF 85mm f/1.8 lens. 85mm has long been a popular focal length for portraiture and should work very well for tight shots of a couple for an engagement shoot.
Just be aware that when mounted on a camera with an APS-C sized sensor the angle of view you will get using an 85mm lens will be more like that of a 135mm lens on a 35mm film camera or full frame digital camera. 135mm has also long been a popular focal length for portraiture using 35mm film.
In practical terms this means you'll need plenty of room to back up away from your subjects to frame waist up or full body portraits. If you are shooting indoors in cramped quarters that won't be possible. If you are shooting outside this will be much less of an issue. However, to do full body or environmental portraits with an 85mm lens might require walkie-talkies to communicate with your subjects!
You'll probably want a wider lens for those types of shots. Something like the EF-S 18-55mm kit lens (not the best, but it works for some folks), or an EF-S 17-55mm f/2.8 (costs a bit more but is worth the difference if you can afford it).
The EF 85mm f/1.8 also has a "cousin" in the EF 100mm f/2. The two lenses are very similar but the 100mm usually performs a bit better optically using the wider apertures than the 85mm at most testing and review sites. If you were using a full frame camera I would wholeheartedly recommend the 100mm over the 85mm. With an APS-C sized sensor, though, you really need to consider the 15-20% more subject distance you would need to get the same framing from the 100mm than from the 85mm. If you need to shoot at 15 feet with the 85mm you would need to shoot from about 18 feet with the 100mm to get the same subject framing. If you are shooting at 25 feet with an 85mm lens you would need to back up to 30 feet with a 10mm lens. Both lenses will allow you to take great photos.