All c. breeze said, all central american cichlids can breed (as long as they don't kill eachother first), almost all south americans can only breed within their genre, however things like kronoheros, mesoheros, and australoheros are closer genetically to central americans and can in theory breed with eachother/central americans. However I have seen accounts of acaras breeding with convicts, but that is unimaginably rare.
As for keeping south americans together, it depends, things like chocolate cichlids, festivums, uaru, geo type things and severums are on the more relatively tame side while things like oscars and acara type things will show more aggression. And then of course things like dwarf cichlids (laetacara, cleithracara, apistogramma, mikrogeophagus, dicrossus) should not be kept with other cichlids as they will probably be killed by things larger than them. Angels and discus I would also not risk putting with other cichlids, and then peacock bass are self explanatory (in case they aren't, don't put them with most other south american cichlids, or any other smaller fish for that matter).
Almost all african cichlids (rift or crater lake/north or west african/madagascan) for the most part cannot cross outside their genus either. While some can, they usually won't, like with julidochromis and rock dwelling neolamprologus from lake tanganyika. Lake victorian cichlids you should not keep together as they can in fact hybridize with most fish in the lake. Some mbuna may cross to outside their genus if they are genetically/visually close enough to eachother like with chindongo and pseudotropheus.
There are only really two ways cichlids will spawn, either substrate spawners that lay their eggs on something and proceed to care for the fry once they hatch, or mouthbrooders, that as the name implies, raise eggs and fry in their mouths. Triggering spawning depends on the species, but for the most part they usually have something to do with shifting water conditions, feeding more, or simulating some seasonal change.
It is not advised to hybridize cichlids purposefully as this reduces your ability to sell them, as most people nowadays will not buy something they cannot identify. If they do however, it then becomes a problem of fish being sold as something they are not, as eventually the original lineage gets lost in time and they end up being sold as a different species, bred to the species they're being sold as, and ruining bloodlines.
Again the ease of breeding also depends on the species itself. Cichlids are the third most diverse group of fish on the planet so there is a lot of variety in terms of how they are bred and the ease of it. The more common mbuna or convicts for example, can be thrown into 20 gallon with mature males and females, and will readily breed on their own if they have optimal water conditions and be pretty hands off. However things like apistogramma, altum angels, or crenicichla will tend to be more hands on in terms of feeding the fry and maintaining water quality. And then things like frontosa, larger central americans, or peacock bass run into the issue of space.
Most tetras on the other hand breed universally the same way. Most are triggered by a surplus of live food, simulated drought, then the introduction of cold rainwater and darkness, scatter their eggs, and don't take care of fry. The ease of which depends on the species however, as some will require you to be more delicate and hands on with the process, and some will again require more specific water parameters that you may need to adjust for, some will spawn more or less readily, etc.
You seem to be a beginner, so if you want cichlids for a 55 gallon I would look into one of the following options:
group of bolivian rams
group of keyhole cichlids (cleithracara maronii)
group or pair of laetacara
group or pair of rainbow cichlids (herotilapia multispinosa)
group or pair of convicts (amatitlania nigrofasciata/siquia)
group or pair of cryptoheros cutteri
group or pair of firemouths (thorichthys meeki)
group or pair of thorichthys maculipinnis
group or pair of neolamprologus brichardi/pulcher
group or pair of julidochromis
group of chindongo saulosi
pair of jewel cichlids (hemichromis lifalili)
group or pair of kribensis (pelvicachromis pulcher)
group or pair of butterfly cichlids (anomalochromis thomasi)
I have color coded them based on where they are from. Green is south american, orange is central, blue is rift lake african, red is west african. The south american options you can keep with most tetras, the central american options can go with swordtails or mollies.