I have the app for Android and I have won against every single Ancient One Multiple times. I almost always win against Yig, Tsathoggua and Cthulhu. I used to almost always win against Azathoth, but I think a recent patch has made that adventure much more difficult. I haven't played Ithaqua enough to know how I stack up against it yet. until Here is the strategy I use (and some reiteration and rebuffing of what others have written):
Some investigators are worse than others. If you want a more challenging game, you could go with the less useful investigators, but it looks like most people don't need this game to be any more difficult.
- Jenny Barnes: My go to investigator. For my style of play, there is absolutely no better character than Barnes. If you play her right and stay ahead of the curve (staying ahead of the curve is the most important part of my strategy, more on that below), Barnes will face every room with both the Red and the Yellow glyphs. She can use her ability every turn: this is a key point for picking investigators, some cool abilities from other investigators turn out to be not particularly useful.
- Joe Diamond: Although rerolls have been lauded as the key to winning, I think they are secondary to having the red and yellow glyphs. Nonetheless, they are still incredibly good and Diamond gives you the ability to reroll a reroll, which nearly doubles your rerolls (you won't use his ability every time)
- I usually use Wilson Richards (from the Cthulhu DLC) and Monterey Jack as the other two investigators, although I play around with the other characters. Richards ability to manipulate the yellow glyph is incredible, you just need to make sure that he always has a common item glyph. Jack gets you tons of unique items, which give you red glyphs, so hopefully you can roll a red glyph every time he enters a room.
My strategy is very aggressive and based upon staying ahead of the curve: the first turn I take with each investigator is pivotal. Like other Lovecraftian games, the odds looked stacked against you: doom tokens seem to multiply only slightly less eagerly than monsters, terror effects pile up, glyphs lock, stamina and sanity drop precipitously. A defensive strategy seems like putting a bandaid on a lost limb, so I go after items like a veteran D&D player. Jenny can turn any item into the 2 glyphs except unique items and she is the most powerful starting out, so I can send her after nearly any items. Joe goes after reroll tokens (the footprint), while Wilson goes after yellow glyphs and Monterey after unique items.
By careful selection of rooms based on their rewards, challenges and prices for failure, and even more careful use of the investigators' abilities and starting items, I should end up with more and/or better items than I began with. I don't worry about picking up Elder Signs too much in the first turn or two of the game. The idea is to get enough items that you can use one or two of them on each investigators' turn, every time. Even if I get behind on the doom track, I can make it up later if I am swimming in items and the endgame usually has at least a couple rooms with two or more Elder Signs.
I try not to use more items than I will gain from a room, unless I am getting multiple Elder Signs. With this strategy,. running out of items is more deadly than taking a hit to health or sanity or (sometimes) failing an adventure. This changes in the endgame. When I am near completion, I can blow the rest of my items getting double or triple Elder Signs from the last rooms.
I don't buy items unless a character's inventory is empty. It's a waste of a turn (the doom clock is ticking) and gold. If you save your gold for the endgame, you can buy the last few Elder Signs you need (cost is 10 gold unless you are playing against Tsathoggua). An investigator with 10 gold or above is a virtual guarantee of an Elder Sign.
I know I wrote that I focus on offense and that defense isn't particularly useful, but that changes with locked glyphs and midnight effects. Get rid of these as soon as you can. Locked glyphs are crippling, nothing will kill your offense faster than losing a die (or more). Use the right investigator to clear out monsters or rooms that lock glyphs: Jenny is great when a green glyph is locked, since she can still use the yellow and red, but you may want to consider another investigator if one or both of the special glyphs are locked.
A few last words: The advice on not hoarding items is sound. Yes, you want a lot of items, but that is so you can use them on every turn. Use the items you have to get more items which in turn you will use to get Elder Signs. Pick investigators who can use their power frequently, preferably at least once every turn. A few monsters aren't that bad (it can sometimes make the game easier if the monster is weaker than the task it replaced, as other posters have pointed out), but once they pile up they can start to make it impossible to clear rooms. Try to keep you monster count no higher than three.
Good luck and I hope this helps!