The image quality from the T5i and the ability to change lenses for different purposes would make it a significant upgrade from your current camera. With respect to the kit lenses, the 18-55 and 55-250 combo would give you a much wider range than the 18-135, but the latter might be a bit better made lens (especially compared to the 18-55). In all three lenses, look for the latest "STM" models, which give better AF performance and in some cases have been improved optically.
Note that STM lenses are particularly useful if you want to be able to shoot video with your DSLR. The focus is designed to be quiet and smooth. (Note, too, that not all Canon cameras support AF in video mode... But I'm a still photographer. I ain't no videographer, so really can't offer much more useful info on this... hopefully someone else can, if it's important to you.)
A more "premium" general purpose lens might be the EF-S 15-85mm IS USM, offering top image quality and the best AF performance, as well as an unusually wide angle of view for this type lens. It would be more expensive than any of the kit lenses, though. In general, USM lenses offer the fastest, most accurate focus.
Other key features are IS or Image Stabilization... which can especially be helpful with longer focal lengths, but is nice to have on any lens.
Canon's L series lenses are their premium quality and among the best made by any manufacturer.
If you like to shoot wildlife, birds, etc., you'll want as much reach with as lon a telephoto as possible. You might even want a 70-300mm, or if you have more to spend and are willing to carry a larger, heavier lens, an EF 100-400L IS or Sigma 120-400 OS.
For landscape, wider is often desirable. The Canon EF-S 10-22mm is a premium lens for this purpose. The Tokina 12-24mm is also a very good lens.
None of the above are macro lenses and you won't have a "macro mode" like you do with your current camera. To shoot high magnification, close-up images you can either add Macro Extension Rings to one or the other of the above lenses, or get a true macro lens. The Canon EF-S 60/2.8 USM is an excellent macro lens. I'm currently trying out a Tamron SP 60/2.0 macro lens... it's a full stop faster than the Canon, so might be better doublng as a short telephoto/portrait lens, but it's AF is slower (so it's not very usable for sports/action shooting).
Canon only sells individual Macro Extension Tubes: 12mm and 25mm. They are high quality, but end up being rather expensive. The Kenko macro ring set is also high quality and not too terribly expensive (about $200). There are cheaper Opteka and Zeikos macro rings ($50 to $75 US) that are more plasticky, but still can work. I'd try to avoid using them with heavier lenses, but they might be fine for lighter ones. (Note: the Zeikos sell under a whole bunch of different brand names - Vivitar, Bower, ProOptic, etc. - they are all the same tho.) Do not waste your money on the really cheap Macro Extension Ring Sets (under $25 US). They don't have the electronic contacts so you lose both AF and control over the lens aperture (there is a work around, but it's a pain in the arse).
There are many ways to set up a lens (and accessory) kit for use on your DSLR. For example, I use a pair of 7Ds (essentially the same sensor as the T5i) and my "premium"/work lens kit is 10-22mm, 24-70/2.8L, 70-200/2.8L IS, 100/2.8 USM macro and 300/2.8L IS with 1.4X and 2X teleconverters. When I need to hike some distance and want to carry a much lighter kit, I more often use 10-22mm (or Tokina 12-24), 28-135 IS USM, Tamron 60/2.0 macro, and 300/4 IS with 1.4X teleconverter. Sometimes I'll carry 70-200/4L IS instead of or in addition to the 300mm. I've been shooting with Canon gear for over 12 years now, so also have a number of other lenses acquired over time for various purposes including more macro choices and some lenses that I use mostly on a full frame 5D Mark II camera. But what works for me might not work for you.
There are many other possibilities for lenses and useful accessories. The beauty of a DSLR is that you can configure it as needed for your particular pusposes. It sounds as if you are interested in expanding your skills as a photographer, so I would highly recommend you pick up a copy of the book "Understanding Exposure" by Bryan Peterson. This might be the best $20 you ever spend on your photography! It is not specific to any particular brand or model of camera, but can help people transition from being a "snapshooter" to being a "real" photographer.
Also consider getting one of the guide books for whatever camera model you end up buying... These sort of pick up where the included user manual leaves off, give you a better overview of the system as a whole to help you build up a kit that meets your particular needs well, and give you a lot of hands-on, real world use examples. Even though I've been shooting for 30+ years and have used a wide variety of cameras, I always make a point of getting a guide book for any new model. I can recommend the guides written by Michael Guncheon, Charlotte Lowrie and David Busch, tho I'd bet there are some other good ones from other authors, too.
For whatever lenses you get, be sure to also get the matching lens hood (comes with some lenses, but a lot of Canon in the price range being discussed it's sold separately). You don't really need "protection" filters. A Circular Polarizer might be really helpful.
You also might want to consider a tripod and an accessory flash, maybe later if not right now. A tripod helps you steady your shots, of course. But it also forces a photographer to slow down, study and think about their shots. So it's a great learning tool, too. The T5i has a built-in, pop-up flash, which might serve for occasional, emergency use. But an accessory flash has a lot of advantages and can be nearly essential for thing such as macro photography and portraiture. An accessory flash is much more powerful and has it's own power supply, so doesn't hasten the drain on the camera's battery. It's also positioned farther from the lens axis, especially if used on a flash bracket with an off-camera-shoe-cord, which helps minimize redeye and ugly shadow effects.
You don't have to get everything right now. But lay out a plan for what you will need to shoot what you want to shoot. Prioritize and start building your kit a bit at a time.
Rainy weather is always tricky. I doubt the T5i is any better or worse than your present camera. Very few point n shoots or DSLRs are truly waterproof. I'd suggest you get some accessories to help protect your camera, lens and all your gear. It can be as simple as plastic bags, gaffer tape and rubber bands... and perhaps a plastic rain poncho that will cover up you and your camera bag. Or there are more sophisticated and tailored rain sleeves.... and even actually watertight underwater cases that allow you to dive to 100 meters or more with your gear. You can spend as little or as much as you wish. Just use some common sense. Cameras are electronic items... essentially specialized computers now... and one drop of water in the wrong place at the wrong time can short them out and ruin your day in a very expensive way! Even tho I live in N. Calif. where it doesn't rain about 9 months of the year, I've been caught out in downpours and had my gear soaked (no harm done... I shut everything down, removed all the batteries and gave things several days to dry out before powering things back up). Now I have camera bags that have waterproof covers built in, a few plastic bags, gaffer tape, rubber bands and OpTech sleeves, as well as some cheap plastic ponchos, in my car and in my camera bags... Cheap insurance, just in case. Even for my gear that's stated to be "weather resistant".
Have fun shopping!
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Alan Myers
San Jose, Calif., USA
"Walk softly and carry a big lens."
GEAR: 5DII, 7D(x2), 50D(x3), some other cameras, various lenses & accessories
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