The advantages you expect to have by making Excel more capable will probably limit what you may want to do with your applications.
But Excel is super flexible, if you don't have large amounts of data, you can do a lot with it. There are workarounds for everything, that can be a good or a bad thing, it depends on who you ask, so let's evaluate that.
Let's say you want to have many users access your workbook, two users are already a lot. You will need to save and close before the next user uses your data. How can you solve that? well, you can export your data to a database system and let Excel become just a front end that retrieves the required data only. That way, any user of the Excel front end will R/W up-to-date data whenever they see fit. Excel can do the reporting just for the user using the data. A database system could do everything though.
So let's say now that you want your data to be in good shape all the time. You want users to enter strings with string columns, numbers in number columns, numbers of a certain length in columns for that type, dates with dates, etc. Clever excel routines and constraints can be put in place. VBA can do wonders here, the worksheet change event can handle all that interaction too. You can add data validation in regular cells and data validation in ListObject cells as well as other kinds of measures. A database system could do everything though, with so much ease. Well, it depends, that will also require forms.
Now let's think for a moment that you expect your data to grow to huge proportions. Power *something* apps can help with that, they can interrogate your data and do some neat stuff with it. In Excel, you can have a lot of data stored, use those tools to get your results. Do keep in mind that sometimes a small amount of data is enough to crash a workbook, if the processing it requires is too much. A lot of Excel workbooks will have a bunch of cells reacting to all sorts of changes, so it's necessary to apply some clever design techniques here. A database system does not need to calculate everything though, you can choose what to calculate, only what's necessary.
How about security? does your data look like it could be sensitive information? Just apply some password and some VBA techniques to keep evil doers from peeking the file. OK, yes, there are a lot of things to remove passwords, bypass VBA protection, etc. You're very exposed in this regard. A database system may offer more security here. It depends on the system though. Don't expect Access to be more secure than Excel. It's not. SQL Server? now we're talking.
Is Excel capable? yes. Is Excel a golden hammer? for many, yes. Can a database system work better than Excel? sure... but it requires some more thought. In the end, what is best? I'd go for learning a database system.
So what if you want to see your data from anywhere, even an iPhone? a web application is better suited for that, but you can employ the central database technique here. You can write from Excel and retrieve the written data from Excel as well. Then, from the smartphone, using a cross-platform technology, you enter a website and the website communicates with the central database and retrieves your data and you'll be able to visualize it from the iPhone. That still requires you to write the web app.
Databases have their advantages over Excel sometimes. But Excel isn't useless either. It can handle smaller tasks fine. With some VBA tricks, it can even do a bit more. But if you're dealing with serious stuff, maybe think about using a database system, it will be harder, but you won't be as limited. Access is a pretty good alternative, if you learn it, it offers a similar user experience to Excel, so it's a great start for someone coming from an Excel background. The only drawbacks of Access are security, visuals and web deployment, you can't deploy an Access app to the internet (but there are workarounds, remote desktops, for example), but Access can communicate with web servers and you can still do a lot of web stuff from it.