Foreflight – this has become the defacto standard for pilot apps. It does almost everything a pilot needs for flight planning, preflight weather and NOTAM checks, in-flight charts and airport information and much, much more. One big downside – it is iOS only. I have an iPad mini that I use exclusively for Foreflight. It can get expensive, but the cheapest subscription ($99/yr as of 2021) is cheaper now than the cost to keep up to date on the Terminal and Sectional charts for LA. Most students will do fine with the cheapest version. I personally use the middle version (Pro – $199/yr), but have students who prefer the $300/yr version.
AviationWeather.gov – I still do most of my weather “quick looks” here. I keep booked marked links that will give me common lists of METARs and TAFs. I can get a quick 
check of PIREPs, METARs, SIGMET/AIRMETs, and radar right off the front page (ensure you have the right things checkmarked) or I can bookmark a version that will show me my local area. With the movable maps, I have to be careful where my cursor or finger is to avoid unexpected scrolling or zooming, but I’ve learned to work around it.
AirNav – I use AirNav to get basic info about airports, such as the runways (numbers, widths, lengths, slopes), frequencies and telephone numbers, pattern altitudes, etc. It includes almost all of the information that would be contained in the Airport Facility Directory, but in an easier to read format, plus more information. It includes sunset/sunrise times, local weather (METAR/TAF), and aerial pictures of most airports. They seem to make their money off listings for FBOs and other local airport businesses. It’s a good place to research FBOs and fuel prices (100LL.com is another option there); the list of businesses is not always exhaustive, but is a good start. Check the reviews for good hints abouts local procedures or restaurants nearby.
SkyVector – I’ve used SkyVector for years to look online at charts. The normal view stitches together the sectional charts, but you can switch into Enroute IFR charts, Terminal Area charts, and many others. They’ve recently updated the site to include more detailed flight planning capabilities, though I’ve used it for basic route planning even before the most recent update.
FltPlan Go – FltpPlan.com publishes an app for both Android and iOS. The app and all the data is free. It includes the ability to view all sorts of charts (TAC, Sectional, IFR enroute) with current location, as well as instrument approach plates and airport diagrams (also with current location). To get the most out of the app, you will need to register on FltPlan.com and log in through the app. The app is not as integrated and polished as apps like ForeFlight, but it works for me and is free. I’ve covered the predecessor to this app before (the non-GO version) .
Lockheed Martin Flight Service – LM is the contractor that now runs the FAAs Flight Service. Call 1-800-WX-BRIEF and you talk to a Lockheed Martin flight briefer. They’ve also created an online service that allows you to get a legal standard briefing that does a good job of walking you through all of the available weather and NOTAM data. If configured correctly, you can file flight plans online and later open and close those flight plans by clicking on links they’ll email or text to your phone. You must register and log in to use the services. Student pilots may have a few hoops to jump through in order to register.
FlightRadar24 – when I send students on solo cross country flights, I feel better if I am able to track where they are and their progress. For many years, FlightAware was my best option, but was much less than 100% in properly showing these VFR flights (and never worked for shorted VFR flights). FlightRadar24’s web site so far seems to have a better success rate (though I don’t yet have much experience with it). Unfortunately, their free app doesn’t allow you to track by tail number; I haven’t tried premium yet to see if it works, but the free web site works well enough for now.
WebTrak – I’ve found the need several times to look at historical flight tracks in my local area. The company that provides WebTrak contracts with many large airports to assist in their noise compliance efforts. As a side benefit they allow tracking of flights near each of those airports in amazing detail and allow you to look back in history. I’ve used this to identify an A380 that flew right over me, to track the movements of a pilot who was clearly in over his head and a danger to others, and to watch flow of traffic that has led to a couple of near miss situations. Not all airports are covered and I suggest googling for a large airport and WebTrak to see what is available. The link I have at the beginning is for Ontario, whose coverage extends to EMT and several other local airports I frequent.
ChartBundle – I like to have paper backups for my local IFR training airports and any time I fly IFR for both my departure, destination, and expected alternates. Chart Bundle allows me to create a PDF of any combination of airports and format it be printed double side so that 4 plates fit on a single page of paper (similar in size to NACO or Jep charts). The web site can be unwieldy and it takes some experimentation to get the right options (I also wish I could exclude some pages that are automatically included), but it’s the best option I’ve found for my paper backups.
Ogimet – if you want historical METAR and TAF data for almost any airport, this is your site. I’ve used this to review past flights for weather trends and to school a couple of pilots with the weather they should have known about on flights they were pushing limits or almost got themselves in trouble.