Completed grid (by @Deusovi):
And as is clearly visible,
the top left is the PORT side (the nautical term for left), so the ship must be going toward the top right.
Across
2. English city shuns Henry Paisley of musical fame (4)
BRAD = English city shuns Henry (BRADFORD - (Henry) FORD) Paisley of musical fame (BRAD (Paisley))
5. Show off expensive top by Prada, moving foot to the left (6)
PARADE = Show off (PARADE) expensive top (E) by Prada, moving foot to the left (PRADA -> PARAD)
7. Battle site's illegal hunter loses track of singer, gets sad (6)
POSADA = Battle site's (POSADA) illegal hunter loses track of singer (POACHER - CHER = POA), gets sad (PO-SAD-A)
8. Capital's rival top blown up (4 4)
PORT VILA = Capital's (PORT VILA, Vanuatu) rival top blown up (RIVAL TOP -> PORT VILA)
9. Our pilot changed fifth course to capital (4 5)
PORT LOUIS = Our pilot changed (PORT LOUI) fifth course (S) to capital (PORT LOUIS, Mauritius)
11. Take dive out of Galician club to city (5)
PORTO = Take dive out of Galician club (DEPORTIVO (de La Coruña) - DIVE = PORTO) to city (PORTO)
12. Make sure striking the right number to follow (5)
ENSUE = Make sure (ENSURE) striking the right number (ENSURE - R = ENSUE) to follow (ENSUE) (5)
15. Domino recalled by the twelfth Greek, a prominent Turk (7)
MUSTAFA = Domino recalled (FATS (Domino) -> STAF) by the twelfth Greek (MU + STAF), a (A) prominent Turk (MUSTAFA (Kemal Atatürk))
16. Capital gets rid of capital bird (4)
ERNE = Capital (BERNE, capital of Switzerland) gets rid of capital (first letter) = bird (ERNE, a type of seabird often seen in crosswords) - Thanks @Deusovi!
17. Rest of list with messy mush making narrow strip (7)
ISTHMUS = Rest of list (IST) with messy mush (HMUS) making narrow strip (ISTHMUS)
19. Clock parts from French and Swiss components (5)
HANDS = Clock parts (HANDS) from French and Swiss components - Thanks @Deusovi!
br>20. Looking back, kitten's father is god of death (3)
MOT = Looking back, kitten's father (TOM -> MOT) is god of death (MOT)
21. Literary David wants Field's metal (6)
COPPER = Literary David ((David) COPPERFIELD) wants Field's metal (COPPERFIELD - FIELD = COPPER)
22. Make a mistake in most of Asia's predecessor to tango (6)
SIERRA = Make a mistake in most of Asia's (SI-ERR-A) predecessor to tango (SIERRA in the NATO alphabet)
23. Chinese cheers began disintegrating in front of medical centre (6)
GANBEI = Chinese cheers (GANBEI) began disintegrating (GANBE) in front of medical centre ((MED)I(CAL))
24. Beggar abandons tavern food item (3)
EGG = Beggar abandons tavern (BEGGAR - BAR = EGG) food item (EGG)
26. Short naval force's star leader takes a step right (3)
USN = Short naval force's (US N(AVY)) star leader takes a step right (SUN -> USN)
27. Writer's horizontal heart and vertical feet flipped (4)
ZOLA = Writer's (ZOLA) horizontal ((HORI)ZO(NTAL)) heart and vertical feet flipped ((VERTIC(AL) -> LA)
30. TV network car for hire front to back (3)
ABC = TV network (ABC) car for hire front to back (CAB -> ABC) - Thanks @chasly from UK!
31. Clarinetist Stiller close to New York (5)
BENNY = Clarinetist (BENNY (Goodman)) Stiller close to New York (BEN (Stiller) + NY)
32. On top of brontosaurus parts (4)
ONTO = On top of (ONTO) brontosaurus parts (brONTOsaurus)
Down
1. Director cuts wood for direction (4)
EAST = Director cuts wood ((Clint) EASTWOOD - WOOD) for direction (EAST)
2. Well done, Italian car (5)
BRAVO = Well done (BRAVO), Italian car ((Fiat) BRAVO)
3. Bone research and development switched for Persian king (6)
RADIUS = Bone (RADIUS) research and development (RAD) switched for Persian king (DARIUS -> RADIUS)
4. Degrade foundation under central idea (6)
DEBASE = Degrade (DEBASE) foundation (BASE) under central idea (I(DE)A + BASE = DEBASE)
5. Capital physical activity of agony moving head back (4-2-5)
PORT OF SPAIN = Capital (PORT OF SPAIN) physical activity (PORT?) of (OF) agony moving head back (PAINS -> SPAIN? Not sure)
6. Artist spied a library in secret (4)
DALI = Artist (DALI) spied a library in secret (SPIE(D A LI)BRARY)
7. Capital's absurd pin-up creator (4-2-6)
PORT-AU-PRINCE = Capital's (PORT AU PRINCE) absurd pin-up creator (PINUPCREATOR -> PORT AU PRINCE)
8. Capital harbour wildly sober in shortest month's limits (4 7)
PORT MORESBY = Capital (PORT MORESBY) harbour (PORT) wildly sober (ORESB) in shortest month's limits (M(A)Y + ORESB = MORESBY))
9. Heartless Polish associated with the upper class (4)
POSH = Heartless Polish (POLISH - LI = POSH) associated with the upper class (POSH)
10. Go around a nutty tree, carve out heart for girl (4)
LEAH = Go around (reverse) a nutty tree (HAZEL), carve out heart (LEZAH - Z = LEAH) for girl (LEAH) - Thanks @Deusovi!
11. A club activity setting (7)
PUTTING = A club activity (PUTTING, as in golf?) setting (PUTTING)
13. Close to divine drink with no heart (4)
NEAR = Close to (NEAR) divine drink with no heart (NECTAR - CT)
14. Perform in reverse obscured by sundown (4)
UNDO = Perform in reverse (UNDO) obscured by sundown - Thanks @Deusovi!
17. Picture a wizard next to a pillar (5)
IMAGE = Picture (IMAGE) a wizard (MAGE) next to a pillar (I looks like a pillar)
18. Emergency signal either way (3)
SOS = Emergency signal (SOS) either way (SOS backwards is SOS) - Thanks @Omega Krypton!
21. Messed-up cube with rights to Ilocano language (7)
CEBUANO = Messed-up cube (CUBE -> CEBU) with rights to Ilocano ((ILOC)ANO) language (CEBUANO)
25. Close, unfinished valley (4)
GLEN = Close, unfinished (GLEN(N) Close) valley (GLEN)
28. Santomean preserve's homeless people skinned (3)
OBO = Santomean preserve's (OBO) Homeless people skinned (HOBO - H) - Thanks @El-Guest!
29. Head down, light brown insect (3)
ANT = 29. Head down, light brown (TAN -> ANT) insect (ANT) (3)
