I took my Pioneer SX-450 Receiver and Pioneer CT-1040W Tape Deck apart to move it. Put it back together. Now can't get a tape to play.

The repair of a dual cassette deck is usually not worth it if you are paying a Technician but since you are doing it yourself, there are screws that must come out of the deck to take the transport out. Some of them require you to take the plastic door off. They are not a treat to work on and often even when you are done they have poor performance anyway. Where the belt goes is indicated on the pictorial diagrams of the service manual. Otherwise it is common sense and a little bit of physics. I have probably done 400 by now. Maybe some TV shops will work on it but they are really not real Technicians for tape decks and most times hacks.

The tape speed is an indication of another problem. Probably the same as all other mechanisms of this type and that is dried up grease and parts that are not moving as they were designed to.
The deck needs to go to a Technician.

You may have some broken tape wrapped around the capstan. The capstan is the small silver post that rotates when the tape is playing. A rubber roller presses the tape against the capstan. This silver post needs to be just that - silver. If it has any brown tape or goop on it, clean off the tape or goop with a small knife. Then follow up with nail polish remover [acetone]. Don't use alcohol because that will harden the rubber roller over time.

These dual cassette decks were throw away units in most cases. The belt inside and the grease in the cam mechanism is in bad shape and the deck can not cycle through the steps it need to so as to reach stop. Only stop mode will allow the door block to release. The tape can safely be extracted by a Technician. The deck need work. Dual cassette decks are not always worn the repair cost.

The deck if in working condition can be plugged into any AC wall outlet or the two AC outlets on the receiver. This is only for power, The Line outs and line in also have to be plugged into the Tape in and record out on the receiver as well. If the CT-F2121 need repairs that is what needs to be done first. Probably a whole set of belts.

Everyone buys dual cassette decks and wonder why they don't last- it needs to see a Technician and if not that it will be landfill.

You have to select any available 'tape loop' .

Receiver Tape Out (Rec) - to INPUT

Receiver Tape In (Play) - from OUTPUT 1 or 2

A tape deck would hang off the back of the EQ using ITS tape loop.

The left side is usually the playback deck and the right side is the record deck.put in a tape on the left side and push play.Then push the record button first and while holding it down ,the play button on the right side deck.One deck will not have record buttons on it,and that is the playback deck.

Appreciate the additional information.

To recap what you need is a receiver that can accommodate:
4 - Pioneer S-DF1-K (15-100W, 8 ohms)
1 - Pioneer PD-F407 25 Disc CD Player
1 - Pioneer CT-300 dual tape deck
1 - Kenwood KD-491F turntable

Possible choices are (keeping in mind that the power output should be less if not equal to 100 watts and a turntable/phono input):
i. Onkyo TX-8522 Stereo Receiver;
ii. Yamaha RX-397 100 Watt Natural Sound AM/FM Stereo Receiver;
iii. Sony STR DE695 AV receiver;
iv. Teac AG-790 200 Watt Stereo Receiver;
v. Denon AVR 888 - AV receiver - 7.1 channel.

Some of the above also includes feature(s) to hook up to your video system. Some may even require a fifth speaker (subwoofer).

Incidentally, you can also choose another brand/model even without a turntable/phono input. As initially posted, a turntable/phone to line level pre-amplifier is only required. Here are some examples:
a. XP200 Turntable Preamp - Nano Series;
b. Pyle® Pro PP999 Phono Preamplifier;
c. Pro-Ject Phono Box Turntable Preamplifier;
d. Radial J33 RIAA Turntable Preamp Direct Box

Good luck with your project.

Better tape decks, like Pioneer, may use a different mechanism for record/play than FF & REW. So it sounds like a belt or mechanical link for the REC/PLAY has failed.