People get infertility tests done in the following 3 cases:

Partner is facing physical problems like irregular menstrual cycles or ovulation (for women), inability to release sperm (for men), etc.
The woman is in her mid-30s or older, and has not been able to get pregnant in over six months.
The woman is in her 20s or early 30s and has been unable to get pregnant in over a year.

Tests to find out the reason for infertility

Tests for both partners

Medical history - A doctor questions the couple about their sex life, birth control methods, use of tobacco, alcohol or drugs, lifestyle, and any history of sexually transmitted diseases.
Physical exam - Physical exam for women includes a pelvic examination and Pap smear test conducted by a gynaecologist. An urologist examines the testes in men.
Blood and urine tests - Tests are conducted to check the levels of various hormones that affect reproductive health of a man and woman. Tests are also conducted to verify the presence of any sexually transmitted diseases.
- Anti-mullerian hormone - Levels of this hormone are tested through a blood test and shed light on a woman's ovarian reserves (egg supply). Levels go down with age.
- Thyroid function test - Problems with thyroid hormone and thyroid-stimulating hormone (TSH) affect a woman's menstrual cycle, specifically ovulation.
- Luteinizing hormone (LH) and progesterone - LH levelsin man can detect any problem with the pituitary gland. In women, LH and progesterone levels can tell if she is ovulating regularly.
- Prolactin - This hormone is produced by the pituitary gland and is used to check for menstrual or ovulation problems.
- Follicle-stimulating hormone (FSH) - FSH levels are sometimes checked to determine ovarian reserve. It may also be used to understand low sperm count in men.
- Testosterone - Levels of testosterone indicate if there is any problem with a man's testicles or pituitary gland. Low levels contribute towards low sperm count.
- Test for sexually transmitted diseases - Urine sample or samples from cervix or urethra are taken to detect any sexually transmitted infection.

Tests for the male partner

Semen analysis - Semen analysis is conducted to determine sperm count (number of sperm), proportion of normal-looking sperms, the number of sperms with normal motility, the number of white blood cells in the sperm, and the amount of sperm.

Tests for the female partner

Pelvic ultrasound - A pelvic ultrasound looks at the state of uterus and ovaries (size and structure) and helps to identify any abnormalities.
Home test for ovulation - A woman can chart her menstrual cycle and understand if and when she ovulates in the month. Over-the-counter home ovulation kits are available that can be used to detect the sharp rise in levels of luteinising hormone (LH) from a urine sample just before ovulation. Ovulation can be predicted by other means too, like charting basal body temperature changes over a cycle (a sharp dip in temperature is seen just before ovulation, and temperature increases soon after ovulation).
Postcoital test - Though not done routinely, this test checks the woman's cervical fluid soon after intercourse. Sample from the cervix is used to see if sperm are alive and able to move normally once inside the cervix.

If the above tests fail to determine the exact cause of infertility, the doctor may ask the woman to undergo more tests:

Hysterosalpingogram (an x-ray test of the uterus and fallopian tubes that can detect any blockage in fallopian tubes.)
Sonohysterogram (ultrasound test using saline to view female reproductive organs.)
Endometrial biopsy (looks for abnormal changes in the uterine lining.)
Laparoscopy (used to look for cysts, scar tissue, fibroids and infection in pelvic organs by guiding a thin, lighted scope through an incision in the abdomen.)

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Sources:

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"Diagnosis & Tests," WebMD.com, https://www.webmd.com/infertility-and-reproduction/guide/infertility-reproduction-diagnosis-tests"Fertility basics," Human Fertilisation and Embryology Authority, HFEA, https://www.hfea.gov.uk/fertility-basics.html"Infertility," MayoClinic.com, Mayo Clinic Staff, https://www.mayoclinic.org/diseases-conditions/infertility/basics/tests-diagnosis/con-20034770"Infertility Tests - Overview," WebMD.com, https://www.webmd.com/infertility-and-reproduction/tc/infertility-tests-overview