GDAL runs on most platforms, including Windows, so I'm not sure where you get the idea it only runs on Macs! The easiest way of getting installed on your machine is to download OSGeo4W, which is an installer for all manner of desktop GIS goodness, from which you can just choose GDAL/OGR.
Once you've done that, you can use the command line tool gdalwarp to reproject the images:
gdalwarp -t_srs EPSG:3857 -s_srs EPSG:9802 -r cubic -co "TILED=YES" input1.tif input2.tif ... input30.tif output.tif
Where EPSG:3857 is shorthand for the "Web Mercator" projection (thanks Sean!), and EPSG:9802 is the shorthand for the Lambert projection. You may need to use a different sampling algorithm instead of cubic, it depends on the sort of data you've got, and how it looks when it is warped. Specifying all the source images in one command line will avoid unsightly seams between the original images. the "TILED=YES" is sensible for almost any geo rasters to speed up access and display. See here for explanation of the parameters.
To split the image into tiles, you can use gdal2tiles.py (which you'll need to install Python, but that can be done with the OSGeo4W installer).
Addendum
It ocurred to me last night that you would need to specify the two lines of parallel and the central meridian of your source images on the command line, just using EPSG:9802 won't cut the mustard if that metadata isn't already in the source images. If that is the case, you'll need to change the -s_srs parameter to something like this:
-s_srs "+proj=lcc +lon_0=110 +lat_1=25 +lat_2=47 +datum=WGS84"
Where +lon_0 is the longitude of the central meridian, +lat_1 is the upper standard parallel, and +lat_2 is the lower standard parallel. If the images you're warping are of the same series, these values are likely to be the same for each image, but it is worth checking them first.
If you're unsure about how these projections work, you could do worse than looking here for an explanation.
Best Answer
Try 3347 as input projection.
https://epsg.io/3347