You do not have to do all this to get songs that are on your computer onto your iPhone, without syncing. In order to do this, you must know where the music files you want on your iPhone are stored on your computer, because you will need to copy them to a Folder (the "Automatically Add to iTunes" folder) within your C:/Users/<username>/Music/iTunes/iTunes Media folder. Make sure that iTunes is set to Not Automatically Sync via the Preferences. Otherwise, when you open iTunes it will delete anything on your iPhone that had not already been synced from your Computer. If you must change this setting first, then after you are done, Close iTunes. If you already know how to copy the files and find the folder/directory (old school term...lol) then do that and then go to the third paragraph when you have completed the copy. Otherwise, go on to the next (2nd) paragraph.
Now, open My Computer and Go to your <Username> folder. Once there, scroll down to your Music folder and open/double click it. Look for the iTunes Folder (open it), and then within it the iTunes Media Folder. This is where you will find the Automatically Add to iTunes folder. On the left side of My Computer you should also see the tiered levels of folders showing like an outline...the Automatically Add to iTunes folder should be visible there as well. Do not close/Hide them. Use this left column to click on and open/view the folder that has the music files you wish to get onto your iPhone. Select these files either one at a time, or all at once. If contiguous/no files in-between them you do not want, then click on the top file, press and hold the SHIFT key and click on the last one. If you have non-contiguous, then press and hold the CTRL key while clicking on each file that you want to copy. Release the SHIFT or CTRL key once all the files are highlighted. Now, right-click on any of the highlighted files and choose Copy from the pop-up menu. Then, on the left where all the folders are visible in the outline format, find the Automatically Add to iTunes folder again. Select it, right-click and choose Paste from the pop-up menu.
Once the copy is finished, all the files should be visible in the Automatically Add to iTunes folder. If you iPhone is currently connected to you computer, eject it. Now, open iTunes, and then reconnect your iPhone it (this is just so we make sure it shows as connected in iTunes). When you open iTunes, it will automatically import those songs you copied. Those files will show in the MUSIC section for iTunes Library...NOT YOUR DEVICE Music section. We still have to copy them within iTunes. I may have a different version of iTunes, so just go and view the Music in the iTunes Library. Find the songs and highlight them...all or individually (use the same SHIFT/CTRL method if you wish). Now, drag them to your iPhone device/device picture on the left. There will be a rectangular marquee as your drag possibly with the number in the corner of the total # of files being copied (the newest version has that). If you look at the top of your iPhone you may see a syncing symbol (two arrows) moving in a circle. The files should now be present on your iPhone with your other music.
If you have an iPod it works differently, and you can transfer Music through the My Computer once the device is connected and recognized. There is a Music folder that you can drag the files/songs to within my computer similarly to how you did it above with the Automatically Add to iTunes folder. Just think of the iPod as a folder/directory...double-clicking and viewing and copying is all the same basic method.
I did the above steps with the new version of iTunes and Windows Vista, as well as over a year ago with an earlier version of iTunes and Vista. Windows 7 (should be basically identical to Vista I believe) and Windows 8 should work similarly, but THIS PC is the My Computer of Windows 8. It sounds lengthy, but that is just with the DETAILS for copying and dragging files/songs around. Other than that is super easy.
