STD's never seem to be a popular topic. Most people are pretty ignorant about them. My main fear is getting Herpes or HIV. With the few partners I've had I didn't ask because I was ignorant. In the future I plan to ask.
The question is when do you ask? If you bring the subject up on a first date then you come across like you just want sex. So do you wait until after you've kissed and fooled around some? That seems reasonable, but then what if he/she says they haven't been tested or you don't entirely trust them? I guess it's reasonable to then wait to go further until after being tested. I think in practice people wait until after sex, then see if the relationship is going to last to go get tested together.
Ideally, everyone that is going to be with a new partner will have a copy of recent test results including blood tests for Herpes 1 & 2 (which they don't routinely do!). If someone showed me that I would feel very comfortable.
I am not too worried about getting a curable STD (syphilis, gonorrhea, chlamydia) because as long as I get tested it can be treated. I am vaccinated for Hepatitis A and B, so I'm not worried about them either.
Hepatitis C (HCV) is not curable, but is not easily transmitted from sexual activity. It is a blood-borne infection so requires blood to blood transmission such as sharing needles, tattoo needles, piercing implements, etc. Sharing other items like toothbrushes and razors can also spread HCV.
HIV doesn't appear to spread easily from oral sex or vagina sex using a condom. So I'd feel comfortable with someone who got tested within the last year and didn't seem too high risk. Of course I'd want them to be tested again if I was with them longer-term.
I'm not sure how much to worry about herpes. In the USA, 20-25% of adults have genital herpes. Condoms only reduce the risk of transmission by 50% because the virus can be on the skin around the genitals. People that have it can still spread it sometimes when they don't have an outbreak. Many people don't even know they carry the virus because they never had a severe outbreak and/or don't realize it's herpes. That's why I tell people to insist on a blood test for herpes when they get tested for other STD's. The tests should only cost $20-30.
Oral herpes (core sores) I'm really not sure about. They say 90% of adults 40+ years old carry herpes simplex type 1 (HSV-1). So if you don't have it, should you ask if someone has it before you kiss them? Some people will say they are disease free, but they get colds sores, so are they really STD-free? And what about the people that don't get cold sores, carry HSV-1 but have never been blood tested so they don't know? They can still transmit HSV-1 with some small probability.