Okay, I'll try to make this one shorter. Once you have your club and your training lots established. Take your household to one of the lots. For example purposes, let's say it's the science lot. Have one sim start making statues on the woodworking table while the other jumps into the observatory and starts building logic (and collecting space prints - good money making source). Remember to have the club set for focused and to buy the handiness and logic skill gain perks for the club. Remember to be running the club, and finally check to make sure the science center has the proper lot traits for training handiness and logic (science lair and homey I think, also your woodworking items will come out better if you also put home studio on the lot). It's a little grindy with your first household, but get one of your sims to 10 Logic and one to 10 handiness using this mechanic. Remember that you don't have to stick to the woodworking table to gain handiness; if your science center has a bathroom (and it should) you can have quite a few things to upgrade there. Once each of your sims is trained to 10 in their respective skill, have them mentor each other in those skills. Have the handy sim teach the logic sim handiness by mentoring him at the woodworking table then at upgrading things (you can even bring them home temporarily and let them upgrade all your stuff at home), then let the sim who was originally handy sit at a chess table and play (alone - if you play with someone, then every time a game ends you have to reset the whole play/mentor sequence) and while he is playing with himself, have the other sim mentor him. In a very short time you have two sims who both have level 10 in handiness and logic.
Now comes the good part. Change your aspirations with both sims to the knowledge aspiration "Nerd Brain". You can always change it back to the original aspiration later without losing progress so don't be shy. Using your new found skills you can easily and quickly conquer the nerd brain aspiration (and train up to 10 or at least very high in rocket science skill at the same time and build and upgrade your science center's rocket). Once you have completed the nerd brain aspiration - you will have enough or near enough aspiration points to buy the fast learner trait. You cannot imagine how much faster your sims train with this perk and if you add the "night owl" and "morning sim" traits as well - you can sleep in the off hours and zip through your training during "morning" and "night" hours.
Now go to the conservatory, change your sims aspirations to the "musical genius" aspiration. Run the club, buy the skill gain perks for the instruments you have chosen, set the club vibe to inspired and set the lot traits to include "great acoustics". Have one sim train on one instrument and one sim train on another (note that the pipe organ is a bad choice of first instrument because there is no club skill gain perk for it). By the time your sims reach level 8 in their respective instruments, they should be ready to start writing those five songs to achieve the third milestone of the "musical genius" aspiration. Writing songs counts as playing music for club purposes and so switch from practicing music to writing music to finish the training (but only if you are up to the milestone in the music aspiration that calls for writing songs - songs written before you reach the milestone don't count). When your sims have each reached level 10 in their given instruments - have them mentor each other in the instruments they have just learned up to level 8. You cannot be mentored while writing songs so for this one you should probably favor writing songs over mentoring after level 8 (achieving the aspiration will bring you closer to aspiration rewards that will really help your sims excel coming out the gate (like the three discussed earlier, but also consider "connections" to boost your career up front and "entrepreneurial" to continue to expedite promotions). Also, a good one to consider is "incredibly friendly" because if you can complete the "friend of the world" aspiration, the "beloved" trait is the bomb.
Look through all your potential aspirations while training. The more aspirations you achieve the more aspiration points you will have to make life just a little easier on your sim. I always try to finish a handful of aspirations before I take a first job with a sim. If you are careful to only spend about half your starter money on your first home and stuff, you can use your public lots for training purposes and have plenty of money to live off of for a few weeks while you train your little pants off. Also consider that a lot of the things you train while doing aspirations bring in a lot of cash (most of my sims never need to work a day in their lives). They can have royalties coming in from books and music and dividends coming in from achieving the "fabulously wealthy" aspiration, they make money while getting the "curator" aspiration by collecting and selling stuff from dig piles, fishing etc. and from (eventually) collecting stuff from Sixam and Selvadorado.