We draw numbers (1-8 depending on # of players). Highest number (1) can either choose the speaker token or draw 3 random race cards and pick 1.  If you take the speaker, you pick races last, if you take the races first, you pick strategy cards last. We play 6-8 player games so we found this to help balance the highest number and the worse. If you take the speaker, all races will have likely been drawn at least once leaving you with the leftovers. 

There's no tremendously satifactory way we've come up with, other than to say we like choosing races with a degree of randomness and not letting people pick whatever they want ( the exception was with the new expansion, since we had 3 people pick one of the new factions, and nobody knew what they did when they picked them.

1.  We have either just done completely random, but that led to some unhappiness, so we went to

2.  Pick one faction at random.  If you don't like it, you get one 'mulligan' but you must accept the second race you choose at random.

#2 has worked out pretty well for us and that's what we've been doing.

In our group, everyone draws 2 races and then selects which of the pair they'd like to use. Occasionally, we'll all just go through the stack and pick out a race that hasn't made it into the rotation in a while, to keep things fresh.

And Ysarril is banned.

 We play with pre-set and agreed upon maps meaning that the group has to come to a more-or-less unanimous consensus on approving the map before we start playing.

Next, everyone secretly writes down their 1st Great Race choice, if there are no duplicates, each player receives the Great Race choice they wrote down.  For players making duplicate choices, they each draw 3 random choices one at a time (returning the unchosen ones to the "pool" before the next player draws), with the last player getting 4 random draws.

Then we randomly assign Home System locations.  The player who gets "Home System 'A' (or "1") is the first speaker and draws the 1st Strategy Card, then we proceed clockwise from there.

 Well, we shall see.  You obviously feel strongly about them not being powerful, but I'm not going to get into a 'theory' war on the internet about it.  I'll say I'm open to considering they are not so great due to number of games I've seen them in.  But they've been excellent from what I have seen. I'll admit that it takes someone who knows what they are doing to play them.   It isn't a good faction for a person learning the game.

We usually just draw entirely at random, everybody gets what they get.  No races are banned.

None of us are overly concerned with nit-picking the mechanical benefits of one race versus another.  We don't fuss over the "quality" of starting fleets and stuff like that.  We just play the game to have fun (and we usually don't actually finish anyway.)

Edit: I do recognize that the Yssaril are probably the most powerful race in the game, but I don't feel the gap is wide enough to stop using them on that principle.  There are enough races to choose from that we don't have to worry about Yssaril being there in every game, spoiling the fun for everyone who isn't them. =P

We pick 2 and choose 1 of those.

After everyone has chosen a race, we randomly distribute the home systems around Mecatol Rex for starting positions.

We then roll a d10 each and highest will be speaker. If 2 or more are tied for highest, reroll among them until one has highest.

Usually we get 2 and chosse one, then rool for speaker - very standard it seems

Then we wanted to try something new

we divided the races into 3 pools, first pool with 3 races, second with 5 or 6 and the rest in the last.

each player would then pick a pool. if they pick pool 1 they get 3 trade goods, pool 2 gives 1 trade good and pool 3 gives 0.

when everyone have chosen a pool we roll to see who gets to pick first. if more people chose the pool, than there are races in it, then the last players to pick move up a pool and pick after the players in that pool, however still gaining the TG's from the pool they joined first.

this rewards players for randomness

we've only actually tried it once - so it might need some tweaks

I take one race token from each race and put it into a bag, shake and each person picks out a token and that's the race they play.

 I take all the home systems and shuffle them.

I then deal them to the players until there is not enough to go around.

Each player picks one and discards the rest.

I like the idea of rolling a d10 and giving the speaker token to the last pick and arranging seating order back up the picking chain so that the first race pick gets last strategy card.

 The thing I like much better about our system (previously posted above) is that I have played a different race every game (12 total games, well, except I have played Fed of Sol twice), and most of our group wants to try new races each time.  Our method more or less ensures that a player who has never played a given race, and wants to try them out, will get the race they have never played.

At least in smaller groups we just choose them. In principle my favorite method, game takes sufficiently long that one shouldn't play something one doesn't actively want. Minor repetition to develop a feel for a race is fine, I may object to more than 3 or so in a row because same cast makes the table feel kinda stale. Personally I currently specifically avoid repeating a race (unless i really misplay/misluck them and want a do-over). Sometimes we even adjust game options used to specific race wishes (recent example: "I want to see fully developed Adv Fighter Naalu on the table for more than a half-turn, lets do Long War".)

For now, races who are practically based on ruining everyone elses fun (and are thus interesting to have at the table occasionally, tiresome to have frequently) like Yssaril or Virus didn't tend to be picked, so we didn't have to adapt the "choosing is fine" method to them.

With new players around, we usually poll them for interests (based on Master of Orion races, gameplay mechanics or setting elements from other games, or whatever source helps) and suggest/assign corresponding races to them. Alternatively, they look through race cards and pick one with the image they find coolest, heh.

Infrequently just did draw 3, choose one, but not recently.

Actually, we have used pretty much all of these, mixed, at the same table before.

The group I play with is all pretty new to the game, so we just pick randomly from the homeworld tiles facedown.

I can imagine going to a "randomly pick one with a mulligan if you don't like the first pick" rule later on as we get more familiar with the different races and their abilities.

I'll  be choosing my races based on their rule sets. My plan is to use this alongside a 40k based idea I have, and the races are to be chosen based on who they're rules are closest to.

for example Sol will be 40k Imperial Guard and Naalu will represent Dark Eldar (hight initiative and rather underhanded).

why would you ban the yssaril? If it is because you belive them to be over-powered, the L1ZIX Mindnet is far more powerful. and otherwise i probably can come up with a reason for how any other (except Xxcha) race is just as potent.

Count Jondi said:

The main reason that most people consider Yssaril to be the most powerful race (by a significant margin) is because of their ability to "skip" actions, which allows them to stall until everyone else has passed and then take whatever actions they have remaining when nobody else can react.  That combined with their unlimited hand size for action cards (allowing them to stockpile "use as an action" cards that they can later burn in between "skips" on a critical turn) can be really powerful.

It's a subtle effect, but it can be devastating in the hands of a player who knows how to use it.  Those who ban the Yssaril obviously feel that this is game breaking. I acknowledge that they are a powerful race, although I've never felt them so powerful that they require being banned.