My girlfriend and I have the Dungeons & Dragons Basic Rules Set 1, Expert Rule Book, and Game Adventure Book. We're looking for free two-player adventures which we could run by ourselves. We'd like them to be simple and straight-forward, as we are beginners. How can I go about finding this kind of adventure?
Go to Drivethrurpg and select the rules system "D&D" and the subsystem "Basic/BECMI" then sort by price (resulting page). This includes several free beginner adventures for B/X D&D but you'll have to search the list to find the ones that meet your needs.
There is also a shared spreadsheet of products released in the OSR (a loosely affiliated set of people who enjoy older RPGs, especially older D&D). This has a type column you can sort or filter to get only adventures. It also includes costs so you can filter for free materials.
I would recommend that you take a look at Black Streams: Solo Heroes. This is a free supplement from Sine Nomine which provides rules allowing you to run a single player character through pretty much any published D&D adventure without requiring modification. This would allow you to look for beginner adventures in general without needing to specifically look for adventures intended to be played one-on-one.
Solo Heroes was originally written for Labyrinth Lord, which is a B/X retroclone, so it should work perfectly with "real" B/X. (I have used Solo Heroes' systems in play, and they work very well, but I haven't personally used them with genuine TSR B/X.)
The primary mechanic of Solo Heroes is based around remapping the damage mechanic so that PCs take hit points of damage, while monsters take hit dice of damage, so it also works well with any version of D&D or D&D-like systems which use HP and HD, not just B/X-based rule sets.
Another option which would allow for solo play of standard modules would be to load up on henchmen. Which option would be better in your case depends on whether you prefer to have a single heroic PC operating mostly-alone (Solo Heroes) or a PC who is a leader, bringing a more traditional party of adventurers into the adventure under their command (henchmen).
Try Scarlet Heroes Quickstart
Sine Nomine expanded Black Streams into a full game known as Scarlet Heroes. It took the basic ideas of the fray die and uneven damage and built a version of D&D around them. It also includes a considerable number of tables and maps to help generate multiple 1-on-1 adventures.
The problem is Scarlet Heroes proper is not free.
There is a free Quickstart Version of Scarlet Heroes that could get you started and see if you want to spend the money. Even if you don't the tools in it, although a subset of the paid rules, should mesh well with your Basic rulebook allowing you to keep going. It will also provide a more extensive set of generate your own adventure tools than Black Streams.
With the Quickstart you could initially run her through the included dungeon "The Tomb of Iron Thorns" to give the rules a shakedown. After that the rules have a step-by-step procedure to guide you through adventure creation of new adventures as well as tables to design treasures.
If you decided you like the ideas in the Quickstart the paid rules include more details for each step, unpopuluated maps, and more tables to create solo adventures.