You need to install SQL Server 2008 first, then before running or rebooting install SQL Server 2008 Service Pack 1.
You are attempting to install the service pack files first, without an existing base installation of SQL Server 2008. Since the Service Pack 1 package does not contain a full installation of SQL Server, you must have already installed the non-service-pack version. The other option would be to create a slip-streamed version of SQL Server 2008 setup by, essentially, combining the Service Pack 1 with the original version on DVD - then installing directly from that. Check https://duckduckgo.com/?q=%22sql+server%22+slipstream+setup for more info.
Don't click "Check for solutions online", click "Run Program" instead. After that completes, install Service Pack 1.
Windows Update will only tell you about service packs for SQL Server, and only when you've enabled the "update other Microsoft products too" option (sorry, the wording has changed across versions, and I don't recall all of the variations).
SQL Server service packs are recommended primarily because the RTM branch (and earlier service pack branches) are ultimately retired and placed out of support. They are also supposedly more reliable and better tested than cumulative updates. I don't know that they should be provided via Windows Update, though, because that just makes it possible for a service pack to be deployed automatically. I don't like that idea at all - and I don't get a warm and fuzzy about how often I have to disable the "force feed me updates" setting and change it to a more appropriate "download them and tell me about them" option.
Cumulative updates, on the other hand, are recommended by some of us (Glenn Berry and I are vocal proponents of keeping up with CUs, because generally we've found them to be just as solid as service packs - though recently there have been a few chinks in that armor). But the message from Microsoft is not so clear. While they do put the updates out and publicize them, each one of them has this text (this is taken from 2012 SP2 CU4):
We recommend that you apply The most recent update release.
The link points to KB 2983249, which gives you links to all of the cumulative updates for that branch. But then, in the CU article, the very next piece of text is:
This cumulative package is intended to correct only the problems that are described in this article. Apply it only to systems that are experiencing these specific problems.
This disclaimer is probably the reason you won't see cumulative updates being pushed through Windows Update. I know they've been working on bolstering the regression testing and therefore the stability that you can expect from cumulative updates, but until that disclaimer disappears, you're going to continue to have to seek out cumulative updates on your own. I try to keep people updated on all updates (service packs, cumulative updates, and out-of-band on-demand hotfixes) for SQL Server 2012 and SQL Server 2014, with some qualitative advice thrown in where appropriate:
From those two posts, you can probably guess that if people had received the most recent cumulative updates automatically, through Windows Update, they would be plenty upset, and justifiably so. I'm all for deploying cumulative updates but this rash of problems has made me adjust my timeline for doing so - like add a month, maybe, so that other people can shake these issues out and I stop being the guinea pig.
Best Answer
SQL Server 2008 R2 is not supported on Windows 10. This means the SQL Server team will not be releasing any security updates or patches if they find any specific issues related to SQL Server 2008 R2 code. There is no way to get updates automatically as none would be released related to Windows 10.
Having said that, SQL Server 2008 R2 installation is not blocked when you try to install it on Windows 10. Of course, it is always better and advisable to install the latest Service Pack (Service Pack 3 is the final SP for 2008 R2).
If you are trying to install SQL Server 2008 R2 for testing purpose on Windows 10, you can go ahead; but since it's not supported it would be of no use to carry forward this idea.
Service Packs are cumulative, so you can directly install SQL Server 2008 R2 SP3 even on RTM bits. There are cumulative updates (CU) released after 2008 R2 SP3 as well, but my advice is don't install CU unless you face specific issues fixed in the CU.