The corridor trails do give you great scenery, but they are packed with people, and the mass campgrounds are not a wilderness experience. So it's great to have the ambition to be more adventurous, but as Dave said – non-corridor trail conditions are immediately far more challenging.
Mid/late April is a great time to be in the Canyon, because it's flower time – the Tonto platform especially is spectacular when the cactus bloom. However, permit applications for April opened at the begnning of December, and you are probably already too late for any popular area. Early May is also a fine time to be there (and will still have flowers), and you have time to get your application in. Don't go any later than mid-May, temperatures quickly become intolerable and downright dangerous in the inner canyon.
I'd agree with the two S Rim loops mentioned as the easiest next choices:
Hermit Trail. You can combine this with a Tonto (West) traverse and Bright Angel to make a loop. It might be sensible to descend BA with the heavy pack, and hike out via Hermit. The Hermit loop is probably the most popular route outside the corridor. Permits will be difficult, and you will always be staying at specific designated campsites with pit toilets. You will have company, but the number of people will still be an order of magnitude less than the corridor trails. Hiking is easy along this part of the Tonto. All the trail is well made and maintained, albeit not of the "highway" quality of the corridors. The hike down Bright Angel is an easy 4.5 miles, so you can certainly get further than the corridor Indian Garden campground for your first night; but bear in mind that the water at the first Tonto campsite, Horn Creek, is to be avoided (heavy metals I think), so you'll just need to carry water the couple of miles from Indian Garden if you camp at Horn. West of Horn Creek, all the other sites along the Tonto there have good water.
The Grandview Trail and the Tonto (East) section between Grandview and S Kaibab is slightly more adventurous, but as mentioned by Dave only the Cremation section is tricky, but I think it's probably within your abilities. On this route, you will have more of a wilderness experience camping "at large" along the Tonto. To complete the loop, you may prefer using the Bright Angel rather than the South Kaibab, because S Kaibab is completely dry and shadeless. A great 4-night itinerary would be to go down Bright Angel, first night at Indian Garden, then camp at Lonetree Canyon (short easy hike downcanyon for good water), Grapevine Canyon (perennial flow at Tonto crossing), Cottonwood (perennial water), out via Grandview. Indian Garden and Cottonwood permits will be in high demand.
The next most difficult trail is probably Tanner. But there's no suitable loop for you: the Escalante Route running West from Tanner is considerably more difficult than the Tonto, and not for an advanced beginner. Then New Hance and Boucher. Boucher is gorgeous, but has sections of "trail" that are almost Class 3 (hands on rock).
Not all of this is immediately necessary for something straightforward like the Hermit Loop, but if you plan to progress to more difficult terrain in the GC, here are some general things to think about:
– Water is always the big issue in the canyon. Always know your next source, and the contingency plan if it's dry. Know the perennial sources. Know when it last rained, to judge whether potholes are likely. Know which canyons give access to the Colorado. If the Colorado will be a water source, be equipped to clear sediment. Be equipped to carry a lot of water if necessary.
– Be as light as possible in non-essential areas, because (a) you need agility on difficult terrain, (b) you may need to carry a lot water. Ditching your tent is the most obvious strategy. It rarely rains, and I camp cowboy style 95% of the time, and I know where the overhangs are for most of the other 5%. If I carry a shelter, it's a 5oz tarp, and never tent pegs – on the rare occasions when I do pitch, it's rarely on a surface that's suitable for pegs, and one thing you can ALWAYS find is rocks!
– Think carefully about the daytime highs and nighttime lows that you will encounter on your planned route and considering the elvations of your planned campsites. There's a huge variation between the rims and the inner canyon. You can go from risk of hypothermia to sunstroke within a few hours. There is very little shade when the sun is high. Even in the hottest weather, I always wear long sleeves and long pants and a wide-brimmed hat. If you get in trouble with heat & lack of water, a good escape strategy may be to rest up when the sun is high and move only early/late or in darkness, so always have an adequate light and batteries. Know what the moon is doing.
– Tricky terrain often involves traversing loose unstable rock. I find that using one hiking pole is the best arrangement. You keep the pole in your downcanyon hand for stability, and the other hand is free to find holds. Wear gloves to protect your hands. Pockmarked limestone is vicious if you grab hold of it to save a slip. I find the Gorilla Grip gloves (disposable, $5 from Home Depot) are ideal, because they have just enough protection, but they are thin enough that you can wear them on the whole time. They will also stop the skin on your hands from painful cracking in the low humidity.
