Your first photo is likely dollarweed, but here are several weeds that look very similar: creeping Charlie  (Glechoma hederacea), pennywort (Centella), dollar weed (Hydrocotyle), and dichondra (Dichondra).  Creeping Charlie has ragged edges, pennywort  and dollar weed leaves connect to the stem in the middle and under the leaf, while dichondra leaves indent where they connect to the stem.  Any broadleaf weed killer should work on these weeds, but they are all hard to kill and several treatments will be likely.  I’ve attached links to two websites.  The first is a listing of all the NCSU resources you will ever need, and more, about weeds and how to control them.  Look especially under weed management for treatment options and Weeds is Turf for control of different weeds. The second gives some good pictures to distinguish the weeds above.  When using any herbicide, read the label directions carefully, as some can only be used on certain types of lawns.

https://gardening.ces.ncsu.edu/weeds/

http://www.walterreeves.com/gardening-q-and-a/creeping-charlie-vs-pennywort-dollarweed-vs-dichondra-identification/

Your other weed is spotted spurge.  This link gives information on control.

https://www.turffiles.ncsu.edu/weeds-in-turf/spotted-spurge/