The PIC24 series is a nice upgrade from the 16s and 18s, without changing too much. Making the switch to PIC32 is a bigger leap, and to ARM is huge. Depends how adventurous you are feeling, and what your goal is. ARM has lots of processing power, but isn't very good for low power devices. For typical microcontroller stuff like controlling things and replacing logic the PIC24 series is probably best.
Generally agreed. If you want to stay with Microchip, for Ethernet you'll need to go PIC32 if you want it integrated. There are options for smaller PICs if you're OK with reduced throughput using their ENC28J60/ENC624J600 devices which have integrated PHY. With most devices including all PICs with integrated Ethernet be aware you will need to add an additional PHY chip. Some TI TM4C ARM devices have integrated PHY, there may be others.
Regarding power consumption, it largely depends what core you're talking about, and how careful you are with clocking not just the core but also all the peripherals. In my experience between PIC32 and ARM M4 at the same clock speeds, the PIC32 has better consumption, however ARM devices tend to have more clocking domains and options (which can also seriously further complicate things!).
Be aware that Microchip are pushing very hard for folk to use their new Harmony framework with PIC32, making the old MAL libraries "legacy". While the MAL is/was rather a hotch potch of APIs, it did work, whereas Harmony, despite being out for over a year, is a real mess in terms both of being buggy, and lacking decent training and documentation. It also does not perform anywhere near as well as the old MAL in many situations mostly due to the unnecessary levels of abstraction. The PDF for Harmony is currently a 5,300 page semi-computer generated PDF.
I would also steer well clear of the PIC32MZ for now which not only forces you into the Harmony framework, but is heavily bug-ridden silicon at the current time with some serious shortcomings in the errata, and even a year later they're finding more and more quite fundamental stuff wrong with them. I can't imagine they characterised much of that chip before releasing it. Stick with the PIC32MX and MAL for now if you go the Microchip 32 bit way.
IMHO, the big plus point about Microchip is the peripherals and their relative simplicity and consistency across all devices. While they're certainly not identical across the board, there's sufficient similarity that going from one device to another doesn't give you too many surprises.
One of the worst things about ARM is knowing where to start when it comes to picking a device from the gazillions of options. What doesn't help is that not only do ARM have a non-obvious naming convention for their cores which their marketing has seen fit to change confusing things further, but the various licensees don't always make it easy to differentiate either.
Having said that, I'd agree regarding the Tiva C launchpads, they're a reasonable enough way to get started. Their EK-TM4C1294XL includes both Ethernet and USB, and has a debugger onboard, all for $20. They also support a system of add on boards like Arduino shields called Booster Packs. TI generally has the best documentation both in terms of written and video: however there's just so much of it, so knowing where to start makes it very intimidating.