Let me start this by saying that I have not done an in depth analysis of all the possible purchase options in the game. In fact, doing so would be impossible as the banner deals that change every couple of days are based on your Shelter Level and VIP Level. The offers you have available at any given time will be different than what other people have available. Also, some of the options in the shop, I just don't have enough disposable income to even consider. But if money is no concern to you, buy whatever the hell you want. What are you doing even reading this? So with that in mind, I will offer what I consider the best purchases for your money (in order).
¶ Irma Starter Pack
This pack is only $5 and comes with way more VIP Points than normal for this price, some good starting equipment, and most importantly: Irma. Irma and the items she comes with will make your early days in Shelter 69 much, much easier for a pretty low price. If you already know you're going to throw a little money to the developers, but you can't afford much - this is the place to do it.
This pack is only available for the first 72 hours after you start the game. However, as good as it is, if you missed it - its not worth starting over for. You can still earn Irma just by playing the game, it will just take you some time and getting to Chapter 3 will be more difficult. Soon after starting Chapter 3, there probably won't be much difference between if you did or didn't buy this offer. By that point you'll have a ton of Champions, including at least one guaranteed Epic Champion, and probably several others from Crates.
¶ Season Pass
These are $10 and assuming you earn enough Daily Points during the season, will get you a Legendary Champion Crate, seven Epic Crates, 700 VIP Points (500 from the purchase, 200 from Seasonal Rewards), and 500 nanites. Compared to buying just a Legendary Champion Crate for $15, this is a good deal.
¶ Chapter Champion Packs
As you progress to new chapters you will get offers for crates that guarantee enough Trust to unlock a new Champion, plus some equipment to go with them. The prices of these are pretty much always under $10 and a pretty decent deal, especially if they offer clothes you want for your Champions. As always, any Champions offered through these deals can still be unlocked through playing. These deals just happen to be much better prices than the random deals you get every couple of days.
¶ Random Banner Ads/Double Deals
These are starting to get into the realm of people with a plenty of disposable money to spend. If your budget is tight, stick to the options above. For one thing, these offers can frequently be misleading. I give the developers the benefit of the doubt here and think they just need to clean up how the offers are presented, and its not that they are intentionally misleading. Especially since sometimes the packs end up being better than you thought (but not usually). The issue here is that the advertisement will pretty much always include the picture of a crate in the ad, but if you look closely you will realize there is no crate in the offer. It's just equipment and some VIP time. Other times it will offer a Rare Champion Crate, but then you buy it and it actually has several crates! In any case, these offers tend to start to get pricey and usually just amount to some equipment and VIP time. Not really worth it if your budget is tight. The one possibility is if the deal happens to offer an outfit you really want but just can't get to drop in game - maybe then it might be worth it to you.
Some Examples:
No! Stop! This deal contains no crates. Only Equipment and VIP Time/Points.
Does it actually say Ashley and Olivia are included? Do you want to spend the money to find out?
¶ Nanites!
These offers are for the whales. Nanites are relatively plentiful - at least plentiful enough that you can get the ones you need for events by just playing the game. Buying nanites (or other resources for that matter) is really only worth it to those who are intending to spend a lot of money and either already bought, or don't want the offers listed above.