Personally, I embark on planes (or near-planar areas with very little Z-difference) so I immediately dig an access and tend to send it even further down for exploration purposes without hesitation.
Unless I'm
very unlucky, I only ever breach caverns from the roof (easy to re-seal, just need one side-mined tile orthogonally and set an up-stairwell to be built by the (usually) in-situ raw mined rock on the spot of the last up-down before the cavern), then send tunnels over a multiple of 16 tiles one one and/or the other axis (to be in keeping with my usual "grand-plan" layout) to where there's a handy chunk of non-cavern rock that I can burrow down, and send a new stairwell shaft down there. Repeat until hit magma. Then deal with the issue of "where the heck is cavern 1?", or whichever, if I've actually managed to luck on transiting through that rough Z-spanning layer apparently through a pillar or non-cavern rock mass. (Just start off another exploratory shaft from above where it should start and see if
that hits it. Repeat until you have.)
All this time, the shafts are probably included in a "priority exploration" burrow, assigned to one of my miners, so that the other (assuming, as I usually do, that there are two in my starting party) can get the non-exploratory infrastructure started. Later on, the exploration burrow may also encompass other test-shafts actively aimed and breaching the caverns (at spots within my current horizon, with the lowest up/down pre-bordered with a standard dig-designation, read for up-stair construction) to get an similarly-safe eyeball on more of the cavern.
By then, I may have received a couple more decent miners through general immigration to buff up my digging squads, and the current fort I have has one miner designated to a burrow in the soil layers intended (eventually) to be subterranean grazing space (to augment the surface pastures that are frankly getting rather crowded, though I'm having to wait for significant moss/etc growth) while the remaining three belong to three separate burrows:
edge-digging duty - defining my planned limit to horizontal expansion across every Z-level
explorartion shafts - basically one tile within the above, featuring the 'perimeter' stairways down the edges, of which one half have now been fully dug to reveal as much cavern as I'd want to know, and the other half is waiting
marble digging - On my accommodation layers featuring significant marble deposits I'm directing the mining of the marble areas as a priority in order to feed my raw marble stockpile beneath my marble-block masonic workshops set to produce marble blocks and a few marble hatches and doors (set to be used only by novice-and-dabbling masons, as I'm trying to use no-quality ones at the moment) in order to fuel some of my OCD-ish building needs.
(All these burrows also include dining and drinking and sleeping areas, BTW, so they can slope off whenever they like.)
The latter one, due to the west->east priority digging, is actually occupying my three multi-burrowed miners all the time they aren't otherwise on a break or refuelling their stomachs/livers/sleep-centres. But I need a lot of marble blocks for my above-ground constructions (ten "shift-cursor"s and one additional tile, corner-to-corner), plus outlying overhead walkways to the marble-block fortifications that I've got (or will have) sentinel creatures sitting in to pick up on ambushes.
Later on, and maybe in small bursts immediately after a siege/ambush has been found and thwarted so as to reduce the possibility of being ambushed while "on the job", I'm going to re-assign the miners (plus woodcutters, where necessary) to strip away even more of the flat plane outside my ground-level fortress limit than I have already removed, especially one and two tiles away from the border (with the ramps removed/negated on the inner part), so that I can finally start strip-mining down virtually everything beyond the limits of my fort (just a few bridge-supports and other pillars), and at least until I get to the stage of re-breaching the first cavern (which by now I have full visual of, in half the map) they can do this without worrying about invaders shooting down at them.
But that's well beyond the scope of the OP's question.
Short answer, get at least one dwarf digging down, and any other miners you have to spare can get your various open areas sorted out. It's more useful to know what's down there than to hold back through excessive caution. Digging smart (and knowing the risks of the unlikely but possible occurrence of breaching a 1Z-high cavern on its wall, and how to quickly isolate that) is
not incompatible with digging quickly.
(Apart from anything else, I have long known that I've essentially got no ores of iron, so early on I started trading early for iron items suitable for melting, which I do at my magma-powered smelters sat just above the magmaducts I sent into the magma sea. I also know that spoiler-metal veins
don't exist within the self-imposed limits of my fortress internals, so I may end up making "external mineshafts" down from the surface and threading their way in-between the various cavern openings towards the likely mineable stuff, if I desire to get the stuff before I actually deign to conquer the caverns by conventionally-armoured forces and walling them up or channel-isolating the bits of them that are "outside". I've had four FBs wander through my various caverns, and three were still in view last time I bothered to look. I'm quite happy for them to stay there, so long as no flying one decided to randomly explore one of my well-shafts. In fact, as is a "humanoid made of fire" and another made of charcoal, I'd quite like them to meet. They're actually in Caverns 1 and 2 respectively, but my visible horizon shows (as well as a lot of burnt flora on the fire-one's level) that there appears to be a cave-route joining the two caverns, so they may end up in the same area.
)